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Suppression of turbulence and transport by sheared flow
P. W. Terry
Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706
The role of stable shear flow in suppressing turbulence and turbulent transport in plasmas and neutral
fluids is reviewed. Localized stable flow shear produces transport barriers whose extensive and highly
successful utilization in fusion devices has made them the primary experimental technique for
reducing and even eliminating the rapid turbulent losses of heat and particles that characterize
fusion-grade plasmas. These transport barriers occur in different plasma regions with disparate
physical properties and in a range of confining configurations, indicating a physical process of unusual
universality. Flow shear suppresses turbulence by speeding up turbulent decorrelation. This is a robust
feature of advection whenever the straining rate of stable mean flow shear exceeds the nonlinear
decorrelation rate. Shear straining lowers correlation lengths in the direction of shear and reduces
turbulent amplitudes. It also disrupts other processes that feed into or result from turbulence,
including the linear instability of important collective modes, the transport-producing correlations
between advecting fluid and advectants, and large-scale spatially connected avalanchelike transport
events. In plasmas, regions of stable flow shear can be externally driven, but most frequently are
created spontaneously in critical transitions between different plasma states. Shear suppression occurs
in hydrodynamics and represents an extension of rapid-distortion theory to a long-time-scale
nonlinear regime in two-dimensional stable shear flow. Examples from hydrodynamics include the
emergence of coherent vortices in decaying two-dimensional Navier-Stokes turbulence and the
reduction of turbulent transport in the stratosphere.
CONTENTS
I. Introduction
II. Phenomenology and Physics
A. Background concepts
1. Basic turbulence properties
2. Confinement of turbulent plasmas
B. Scaling theory for suppression of turbulence
1. Dimensional analysis
2. Relative motion of fluid parcels
3. Asymptotic analysis
C. Role of flow shear in linear instabilities
D. Observations of transport barriers in plasmas
E. Experimental and numerical tests
III. Related Hydrodynamic Phenomena and Concepts
A. Stable shear flow
B. Rapid-distortion theory
C. Geostrophic flows and other examples
IV. Generation of Sheared E⫻B Flow
A. Radial force balance
B. Mean ion flows
1. Reynolds stress
2. External biasing
3. Injected radio-frequency waves
4. Orbit loss
5. Nonsymmetric transport
C. Miscellaneous processes
V. Suppression in Toroidal Plasmas
A. Plasma geometry and magnetic topology
1. Shear flow stability with magnetic shear
2. Instabilities in a nonuniform magnetic field
3. Shear straining in a torus
B. Effect of flow shear on transport fluxes
VI. Flow Shear and Enhanced Plasma Confinement
A. Tokamak
1. Spontaneous edge transport barrier
2. Externally induced edge transport barrier
3. Spontaneous internal transport barriers
4. Externally induced internal transport barrier
B. Stellarator
Reviews of Modern Physics, Vol. 72, No. 1, January 2000
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C. Reversed-field pinch
D. Linear magnetic configurations
VII. Transition to Enhanced-Confinement States
A. Bifurcation of plasma state
B. Transition modeling
1. Two-step transition models
2. First-order critical transition theory
3. Second-order critical transition theory
C. Transient phenomena
VIII. Generalization to Related Systems
A. Coherent vortices in Navier-Stokes turbulence
B. Transport of stratospheric constituents
1. Meridional transport across zonal flows
2. ␤-plane model of meridional transport
C. Shear flow in self-organized criticality
IX. Conclusions
A. Summary
B. Open questions
C. Future directions
Acknowledgments
References
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I. INTRODUCTION
Turbulent transport can be greatly reduced in the
presence of stable shear flow. The mechanism (Biglari,
Diamond, and Terry, 1990) is quite simple. When a fluid
eddy is placed in a stable laminar background flow
whose speed varies transverse to the flow direction, the
eddy is stretched and distorted as different fluid parcels
in the eddy are advected (carried along) at different
speeds. If the eddy is isolated, it can be stretched to
many times its original scale length. When the eddy is
part of a turbulent flow, however, it loses coherence
when stretched to the eddy coherence length along the
direction of the background flow (see Fig. 1). The eddy
coherence length is the distance over which the eddy
flow remains correlated and can be thought of as
roughly the distance between two adjacent eddies of
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©2000 The American Physical Society
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P. W. Terry: Suppression of turbulence by sheared flow
FIG. 1. A reference eddy [(a), no shear flow] sheared by unidirectional plane shearing (b) with u x (y)⫽ ␣ y. If the eddy is
isolated it stretches into the shape indicated by the gray shaded
curve. In turbulence, the eddy loses coherence in a coherence
length, represented as a breakup into two eddies. The loss of
coherence reduces the y scale relative to that of the reference
eddy.
comparable scale, a distance on the order of the eddy
diameter in fully developed turbulence. Fluid parcels
that move an eddy coherence length become subjected
to the advecting flows of other eddies and are no longer
identifiable with their original eddy motion. In the absence of the background shear flow, the time scale for
this loss of coherence defines the eddy lifetime. Dimensionally, the eddy lifetime is the eddy rotation period,
commonly called the eddy turnover time.
In the presence of a background shear flow whose rate
of differential advection exceeds the eddy turnover rate,
eddies stretch to a flow-wise eddy coherence length in a
fraction of the time they would normally take to turn
over were there no shear. Consequently, the eddy lifetime is shortened. Assuming turbulence whose driving
source is unaffected by flow shear (such a source might
be external stirring, or an instability associated, for example, with a thermal gradient), the decrease in the correlation time implies a decrease in turbulent intensity
(eddy velocity). This follows because the rate of turbulent energy dissipation, given roughly by the turbulent
energy divided by the correlation time, temporarily exceeds the forcing rate when the correlation time is reduced. This leads to a transient decay of turbulent energy until a new balance is established with lower
energy. Also, because of the rapid flow-wise stretching
(along the direction of the flow), fluid parcels traverse
only a fraction of the original eddy diameter in the
shear-wise direction (across the flow in the direction of
the gradient of flow speed) before the eddy loses coherence. Eddy scales in the shear-wise direction are thus
reduced as a consequence of the background shear flow.
If an advected scalar is present, its rate of turbulent
transport across the background shear flow is also reduced. This follows because the turbulent intensity and
shear-wise eddy coherence length are reduced, thus reducing the speed and step size of a random-walk transport process.
In magnetically confined fusion plasmas, the reduction
of cross-flow transport by shear flow is often localized to
a region identifiable as a transport barrier. Such barriers
are now widely utilized in fusion plasmas. Before the
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
discovery of the first robust and reproducible transport
barrier by Wagner et al. (1982), plasma confinement and
fusion energy production were seriously limited by
small-scale turbulent fluctuations. These had long
seemed irreducible, because they are driven by the steep
gradients of temperature and density needed to confine
and insulate hot fusion plasmas away from material surfaces. With shear suppression, the heat loss caused by
these fluctuations has been controlled for the first time.
Recent results underscore the impact of shear suppression. Record values of confinement time, fusion power
output, fusion neutron yield, and the ratio of fusion
power output to input heating power have been
achieved using flow-shear-induced transport barriers in
the plasma edge [in the hot ion H mode of the Joint
European Torus (JET), Gibson et al., 1998], or in plasmas inferred to have a large sheared flow and significant
suppression of turbulence (in the supershot of the Toroidal Fusion Test Reactor; Ernst et al., 1998). In other
experiments shear suppression in the plasma core has
led to the reduction of transport in tokamak plasmas by
orders of magnitude to the minimum level set by thermal, collisional motion (Levinton et al., 1995; Strait
et al., 1995). A variety of measurements have confirmed
the role of flow shear in impeding the transport and diminishing the intensity of turbulence.
This review is intended to present the basic physics of
flow-shear-induced transport barriers, to outline the extensive developments in fusion plasma physics pertaining to such barriers, to examine related phenomena in
hydrodynamics, and to explore wider applications of
these ideas. Prior reviews1 have been written for specialists and have given little or no discussion of applications
outside the subject of magnetically confined plasmas.
Moreover, whereas prior reviews have emphasized experimental phenomenology (Groebner, 1993; Burrell,
1994, 1997; Moyer et al., 1995) or the theory of electric
fields in plasmas as applied to collisional transport (Itoh
and Itoh, 1996; Ida, 1998), this review treats turbulence
as a fundamental feature of the systems of interest and
develops key theoretical concepts of relevance to experimental phenomena. The level of presentation assumes
expertise in neither plasma physics nor turbulence, but a
knowledge of general physics at the level of a practitioner in any physics specialty area. This review emphasizes
the striking robustness and universal character of flowshear-induced transport reduction in plasmas. As described in Sec. VI, flow shear reduces transport and turbulence in a variety of magnetic confinement
configurations and in different regions of the plasma
ranging from the hot core to the cold edge. In contrast,
the fluctuations themselves lack such universality,
changing character from device to device (Liewer, 1985;
Wootton et al., 1990) and within any given device in different regions of the plasma (Durst et al., 1993). The
1
These include Stambaugh et al., 1990; Groebner, 1993; Burrell, 1994, 1997; Moyer et al., 1995; Itoh and Itoh, 1996; Carreras, 1997; Ida, 1998.
P. W. Terry: Suppression of turbulence by sheared flow
universality of shear suppression in plasmas and the dynamical similarities between plasmas and ordinary nonionized fluids strongly suggest that shear suppression
should occur in nonionized fluids and in other types of
dynamic media. Section VIII provides a few examples
that have been examined to date. These examples are
intended to be suggestive, not exhaustive, and hopefully
will stimulate further exploration of this topic. The review is structured so that Sec. II provides a selfcontained, brief overview of the basic physics and phenomenology, with development, detail, and applications
reserved for the remaining sections.
After a brief presentation of underlying concepts, Sec.
II describes the basic scaling theory for shear suppression (Biglari, Diamond, and Terry, 1990), using both
simple dimensional arguments and more rigorous mathematical analyses. Collective instabilities in plasmas can
also be stabilized by flow shear (that is not itself unstable), as illustrated by a heuristic criterion (Hassam,
1991; Waltz, Kerbel, and Milovich, 1994). Turning to experiment, we describe the basic features of the type of
barrier first observed and studied in plasmas (Wagner
et al., 1982). The presence of flow shear was not initially
detected, but the features that were observed, specifically, a localized steepening of gradients of density and
temperature, and improved confinement, indicated a localized barrier to transport. Later measurements, including those of fluctuation levels, transport fluxes, and spatial profiles of mean flows, confirmed that flow shear
suppresses turbulence and transport in this type of barrier. Specific experimental tests of the theory are described by Ritz et al. (1990), La Haye et al. (1995), and
Moyer et al. (1995). These include measurements of the
correlation time, shear-wise correlation length, amplitude reduction, and the criterion for significant reduction, and a comparison with the predictions of the scaling theory.
Suppression of turbulence and transport by flow shear
occurs in nonionized fluids but is not a familiar hydrodynamic phenomenon. The reason stems from three additional requirements for suppression, beyond the criterion that the shearing rate exceed the eddy turnover
rate. These requirements are routinely met in fusion
plasmas, but are difficult to satisfy in nonionized fluids.
They stipulate that the shear flow must be stable, that
turbulence must remain in the domain of flow shear for
longer than an eddy turnover time, and that the dynamics should be two dimensional. In nonionized fluids,
shear flows are typically unstable, shear is often present
only for a short time in the frame of the flow, and the
dynamics are usually three dimensional. As discussed in
Sec. III, under these conditions turbulence is driven by
shear instead of suppressed, is advected into and out of
the region of shear before the nonlinearity can decorrelate fluctuations, or its vorticity may be amplified. When
turbulence is advected through a region of strong flow
shear in a time shorter than the eddy turnover time,
rapid-distortion theory, a widely used technique in hydrodynamics, can be applied to trace out fluid motions
(for recent reviews see Savill, 1987; Hunt and CarruthRev. Mod. Phys., Vol. 72, No. 1, January 2000
111
ers, 1990; Hunt, Carruthers, and Fung, 1991). In Sec.
III. B, rapid-distortion theory is shown to be the shorttime, linear counterpart of the long-time, nonlinear scaling theory described in Sec. II. B. Section III. C introduces stratospheric geostrophic flow, a type of
hydrodynamic flow that satisfies the three requirements
stated above. (Geostrophic flows are flows whose time
scale exceeds the planetary rotation rate.) In simulations, suppression of geostrophic turbulence in the
strong-shear regions of a stable, large-scale jet has been
noted, although the mechanism was not elucidated
(Shepherd, 1987). Other hydrodynamic examples that
relate to the scaling theory and underlying physics are
also briefly discussed in Sec. III. C.
In plasma transport barriers, the flow shear is produced by forces in the plasma that respond both to conditions in the plasma and to external forces. The creation of transport barriers in plasmas thus involves the
suppression of turbulence by a given shear flow (as described in Sec. II) and the generation of flow. These two
processes can be linked in complicated ways. The physics of flow generation in plasmas is presented in Sec. IV
by examining momentum balances for plasma flows and
the individual forces that contribute to the balances. The
flow responsible for plasma transport barriers is the E
⫻B drift, a motion common to all plasma charge species, so named because its magnitude and direction are
proportional to the cross product of the local electric
and magnetic fields. Section IV describes how this flow
is affected by the equilibrium ion pressure, ion rotation
rates, turbulent Reynolds stresses (Carreras, Lynch, and
Garcia, 1991; Diamond and Kim, 1991), externally manipulated electric fields (Taylor et al., 1989), preferential
loss of a plasma charge species (Itoh and Itoh, 1988;
Shaing and Crume, 1989), and anisotropies in transport
fluxes (Hassam et al., 1991). The relationships between
flow-shear-induced suppression and flow generation are
developed in Sec. VII.
The basic theoretical notion of flow-shear-induced
transport reduction was formulated for a straight plasma
in a uniform magnetic field. In Sec. V the theory is extended to account for the complications of toroidal
plasma confinement geometry and magnetic-field inhomogeneity. The toroidal configuration and the spatial
variation of the confining magnetic field impart a particular structure to the turbulence and to plasma flows.
Consequently, the scaling theory of Sec. II continues to
serve as a paradigm for the effect of flow shear in turbulent plasmas, but the specifics of its predictions are
modified for toroidal geometry (Hahm and Burrell,
1995).
A critical feature of the spatial variation of the magnetic field in magnetic confinement systems is magnetic
shear, a measure of the degree of field-line twist at different locations within the plasma. Typically magnetic
shear localizes fluctuations to the vicinity of special surfaces within the plasma. This property imposes a constraint on the way in which flow shear affects fluctuations (Carreras et al., 1992). Magnetic shear likewise
affects the stability of sheared flows, significantly raising
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P. W. Terry: Suppression of turbulence by sheared flow
the threshold of basic flow-shear-driven instabilities such
as the Kelvin-Helmholtz instability (Chiueh et al., 1986).
In Sec. V. B, the effect of flow shear on the fluctuation
correlations that underlie transport is introduced in the
context of a simple model that includes magnetic shear
(Ware et al., 1996, 1998). This question is revisited in
Sec. VIII. C, where the effect of flow shear on transport
in systems with self-organized dynamics is examined
(Diamond and Hahm, 1995).
Since the first observation of transport barriers in
magnetically confined plasmas, significant progress has
been made in controlling and manipulating these barriers. These efforts are reviewed in Sec. VI. Historically,
flow-shear-induced transport barriers were first produced spontaneously in the edge of a type of toroidal
plasma known as a tokamak (Wagner et al., 1982). Later
it was found that the edge flow shear region in such
barriers could be externally induced by charging the
plasma edge with a biased probe (Taylor et al., 1989). It
was also discovered that under certain conditions, the
edge flow shear region extended itself inward toward the
center of the plasma, producing a core transport barrier
(Jackson et al., 1991). Recently, transport barriers have
been initiated in the core, independently of edge barrier
formation (Levinton et al., 1995; Strait et al., 1995;
Kimura et al., 1996; Mazzucato et al., 1996). In these
plasmas, the turbulent transport is reduced to the point
where loss rates are governed by particle collisions. The
spontaneous core barriers are often produced in conjunction with certain magnetic-shear configurations, but
otherwise exhibit common features of edge transitions.
It has been predicted that core barriers can also be induced externally using injected rf waves (Craddock and
Diamond, 1991). Experimental efforts to induce core
barriers in this fashion have been limited and have produced mixed results (LeBlanc et al., 1995; 1999). The
above results pertain to the tokamak, the most developed magnetic confinement device. Alternative magnetic confinement approaches to the tokamak have also
achieved enhanced confinement operation in connection
with the formation of regions of strong flow shear. Work
done in stellarator, reversed-field pinch (RFP), tandem
mirror, and Z-pinch configurations is reviewed in Secs.
VI.B–D.
Flow-shear-induced transport barriers in plasmas are
created as part of a transition in which mean quantities
transiently undergo an adjustment to new values in response to an internally or externally driven change
within the plasma. Prior to the transition, the E⫻B flow
is not strongly sheared, and the fluctuations are large.
After the transition there is a large shear in the E⫻B
flow, and the fluctuations have diminished. Other quantities involved in momentum balances, such as ion rotation rates, can also change dramatically at the transition.
Because the plasma state undergoes a fundamental
change in properties as it passes through the transition,
the transition is generally labeled as a bifurcation process. Section VII presents transition phenomenology as
observed in experiment and reviews theoretical models
of the transition. The theoretical models generally fall
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
into two categories, dictated in part by which flow-sheargenerating force is treated as dominant, an issue not yet
fully resolved. In the first step of two-step transition
theories (Itoh and Itoh, 1988; Shaing et al., 1990), conditions within the plasma or an external force (such as the
force produced by an electrically biased probe), charge
up the plasma and create a sheared E⫻B flow. In the
second step, turbulence and transport are reduced in response to the flow shear. In single-step transitions (Hinton, 1991; Hinton and Staebler, 1993; Diamond et al.,
1994; Terry et al., 1994), flow generation and turbulence
suppression evolve as integrated elements of the transition and cannot be separated. Single-step theories are
analogous to first- and second-order phase transitions in
continuous media. Mathematically, the plasma states before and after transition are fixed points of turbulent
fluid closures that model the key forces involved in the
transition.
Section VIII is devoted to the application of flowshear-induced transport reduction to nonplasma systems. Efforts in this area are in their infancy. Section
VIII. A reviews work that shows that certain vorticity
fluctuations in two-dimensional Navier-Stokes turbulence become coherent (McWilliams, 1984) as a result of
the strong flow shear in their periphery (Terry, 1989;
Terry et al., 1992). This flow shear suppresses ambient
turbulence, composed of the eddy motions of loweramplitude vorticity, enabling the vortex to escape being
mixed. For a vorticity fluctuation to become coherent it
must exceed an amplitude threshold. The threshold is
consistent with the evolution of the vorticity probability
distribution function in spatially intermittent turbulence,
and with the spatial structure of coherent vortices in the
simulations. Flow shear has also been speculated to play
a role in atmospheric and oceanic transport barriers, as
described in Sec. VIII. B. Transport barriers in the
stratosphere have been detected through satellite observations and in situ measurements, and have generally
been attributed to other mechanisms (McIntyre and
Palmer, 1984). Theoretical and numerical modeling of
the transport of atmospheric constituents in twodimensional stratospheric turbulence (Ware et al., 1995,
1999) indicates that there is suppression of turbulent
transport by the flow shear associated with polar jets and
zonal flow (flows along a fixed latitude). The role this
process plays in observed transport barriers, which involve other complex geophysical processes, is an open
question. Section VIII. C examines dynamical media in
which transport is governed by a process with selforganized criticality (Bak, Tang, and Wiesenfield, 1987).
The presence of wind shear incident on sand pile automatons suppresses the excitation of large-scale avalanchelike transport events (Diamond and Hahm, 1995;
Newman et al., 1996).
The discussion of Sec. VIII. C completes an elucidation of four distinct effects of stable flow shear on fluctuations. These are (1) the fundamental process of direct
reduction of scales and amplitude in turbulence (Sec.
II. B), (2) the stabilization of collective modes that are
otherwise linearly unstable (Sec. V. A. 2), (3) the disrup-
P. W. Terry: Suppression of turbulence by sheared flow
tion of correlation between the fluctuations of an advectant and the advecting flow, resulting in a reduction
of transport beyond that accounted for by decreased
fluctuation amplitudes (Sec. V. B), and (4) the disruption of the extended correlation of large-scale avalanchelike transport events (Sec. VIII. C). This review
concludes with a brief summary in Sec. IX, followed by a
discussion of open questions and current research directions.
II. PHENOMENOLOGY AND PHYSICS
A. Background concepts
1. Basic turbulence properties
Flow shear affects turbulence because it modifies the
time required for correlated fluid motion of a given scale
to lose coherence. The loss of correlation arises from the
nonlinearity of turbulent advection, which acts both to
scramble fluid motion and to spread its energy from one
scale to another. This process can be readily appreciated
from the Navier-Stokes equation,
冋
册
⳵u
1
⫹ 共 u•ⵜ 兲 u ⫽⫺ ⵜp⫹ ␮ ⵜ 2 u,
⳵t
␳
(2.1)
where ␳ is the mass density, u is the flow field, p is the
pressure, and ␮ is the kinematic viscosity. For incompressible flow (ⵜ•u⫽0) the Laplacian of the pressure is
related to the divergence of the advective term
兵 ␳ ⫺1 ⵜ 2 p⫽ⵜ• 关 (u•ⵜ)u兴 其 and therefore can be incorporated into the advective derivative (Kraichnan, 1959).
The evolution of the flow is governed by two forces,
each with a characteristic time scale. These are the inertial force, represented by the advective term (with pressure included through the identity just mentioned), and
the dissipative force, represented by the viscous term.
The inertial force describes self-advection of the flow.
From the second term of Eq. (2.1), the time scale of the
inertial force is
1 u
⫽ ,
␶e l
(2.2)
where u and l are characteristic flow and length scales.
Turbulence consists of a hierarchy of such scales. Because turbulent flow possesses vorticity, ␻⫽ⵜ⫻u, it is
useful to think of the flow as a hierarchy of eddies of
different scales. Consequently, if u is the characteristic
flow velocity of eddies of scale l, ␶ e is the time scale of
that eddy. At each position, the flow is a superposition
of many eddies of different scales all contributing to the
advection process. The flow is random because the
transfer of energy between scales is constantly changing
the configuration of eddies. Motion at a given scale loses
coherence, i.e., any given eddy decays, due to advection
by other eddies. The decay time is ␶ e because that is the
unique characteristic time of the advection process. This
time is called the eddy turnover time. By the time an
eddy rotates, its energy has been transferred to other
eddies and it has decayed. The energy transfer process is
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
113
conservative, i.e., in the absence of viscosity, advection
causes no loss of energy, only transfer of energy from
motion on one scale to motion on another. The invariance of energy leads to self-similar spectral energy transfer. The turbulent flow velocity at each scale adjusts itself so that the energy transfer rate u 2 / ␶ e is invariant for
all scales in which viscosity has a negligible effect on the
dynamics (Kolmogorov, 1941).
The effect of viscosity is quantified by the viscous dissipation rate, ␶ d⫺1 ⫽ ␮ /l 2 , the rate at which the energy of
fluid motion is dissipated and converted to heat. Because viscous dissipation is diffusive (giving the l 2 factor
in the denominator of ␶ d⫺1 ), the dissipation rate increases at smaller scales faster than the turbulent decorrelation (eddy turnover) rate. Therefore, at large
scales, ␶ d⫺1 ⬍ ␶ ⫺1
e , and energy is transferred between
scales with negligible dissipation, establishing the selfsimilar energy cascade. At smaller scales, ␶ d⫺1 ⬎ ␶ ⫺1
e , and
energy is dissipated before advection can pass it to other
scales, terminating the cascade. In three-dimensional
(3D) Navier-Stokes turbulence, energy injected at large
scale, where ␶ d / ␶ e Ⰷ1, is transferred to successively
smaller scales by the cascade. This process continues until reaching the Kolmogorov scale at which ␶ d ⫽ ␶ e ,
whereupon the cascade ceases and energy is converted
to heat. The ratio of ␶ d / ␶ e for the flow U at the largest
scale L defines the Reynolds number Re⫽UL/␮, which
for turbulent motion is necessarily much larger than
unity. Scales between L and the Kolmogorov scale are
referred to as the inertial range. The turbulence considered in this review is dominated by the inertial force,
i.e., ␶ d / ␶ e ⬎1, so that the decorrelation rate is governed
by turbulent advection and is given by the eddy turnover
rate, Eq. (2.2).
2. Confinement of turbulent plasmas
In magnetically confined fusion plasmas it is necessary
to contain and isolate from material surfaces ionized
gases whose temperatures and densities exceed 10 keV
(10 8 K) and 1013 particles/cm3. Moreover, confinement
must be of sufficient duration for fusion interactions to
replenish the heat lost from neutron fluxes, radiation,
and transport. In most devices, the plasma is toroidal.
By design, the current and magnetic field lie on nested
tori within the plasma called magnetic-flux surfaces. Together, the current and magnetic field exert a bulk force
on the plasma given by their vector cross product. This
force is normal to the flux surfaces and can therefore
contain the pressure of the plasma, as indicated in Fig. 2.
For simplicity of presentation we treat only the case for
which the flux surfaces have circular cross sections and
are concentric. In this case, the normal direction is radially outward. The balance of magnetic force and pressure specifies a steady state of the mean ideal magnetohydrodynamics (MHD) equations (Freidberg, 1982).
Experience has established that ideal MHD adequately
reflects the force balance as measured in plasmas, with
the magnetic field determined from the current density
through Ampère’s law. The mean states of ideal MHD
114
P. W. Terry: Suppression of turbulence by sheared flow
On the slower nonideal time scale, the pressure gradient and other inhomogeneities of the equilibrium drive
slow-time-scale turbulent fluctuations. Together with
two-body collisions, these fluctuations produce a flux of
heat and particles out of the plasma. A class of slowtime-scale fluctuations characterized by spatial scales
that are small compared to the dimensions of the plasma
generally dominates the losses of heat and particles
(Liewer, 1985; Wootton et al., 1990). Although it has not
been possible to attribute the dominant losses to fluctuations associated with a specific collective plasma mode,
measurements have confirmed that the losses are due to
fluctuations and not collisions (for example, see Ritz
et al., 1989). In some cases fluctuation-driven loss rates
exceed those due to collisions by orders of magnitude.
Energy and particle loss rates are measured and employed as a figure of merit for confinement. In steady
state, the energy loss rate is balanced by injected power,
and the energy confinement time ␶ E is defined as the
ratio of thermal energy stored in the plasma to input
power P:
FIG. 2. Flux surfaces of a magnetically confined toroidal
plasma. In (a), the currents and magnetic-field lines lie on
nested tori. The J⫻B force balances the plasma pressure. In
(b), simple toroidal coordinates used for flux surfaces with circular cross section are shown.
are often referred to as equilibria. These are not equilibria in the thermodynamic sense, and they are steady
only over the short time scales of plasma motion for
which the resistivity is negligible, i.e., for motions in
which the magnetic field is carried by the plasma with
negligible resistive diffusion (Jackson, 1975). Over
longer time scales, slower processes, such as two-body
collisions and resistive instabilities, cause the plasma to
evolve from one ideal equilibrium to another and set the
radial variation of pressure and current.
For the plasma to remain stationary on the ideal time
scale, the equilibrium must be stable to collective motions whose evolution occurs on the ideal time scale.
Stable, confining equilibria generally require that the
magnetic field possess helicity, i.e., that it have components in both toroidal (the long way around the torus)
and poloidal directions (the short way around). These
directions are indicated in Fig. 2. It is also common for
the field-line pitch, or the ratio of the toroidal component to the poloidal component, to vary in specified
ways from one toroidal surface within the plasma to the
next. A magnetic field with this radial variation is said to
have magnetic shear. The magnetic field can be created
by currents in external coils, which themselves must be
helical if this is the sole source of the magnetic field, or
by a combination of external windings and internal
plasma current. In the most common toroidal configuration, known as the tokamak, external windings produce
the toroidal field, and plasma current flowing toroidally
produces the poloidal field. The toroidal field typically
exceeds the poloidal field by an order of magnitude.
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
␶ E⫽
3
2
冕
n o 共 T i ⫹T e 兲 d 3 x
P
,
(2.3)
where T i and T e are ion and electron temperatures multiplied by the Boltzmann constant, and a neutral plasma
with equal ion and electron number densities n o is assumed. The energy confinement time ␶ E is global, i.e., it
is sensitive to losses occurring throughout the plasma.
Physically, ␶ E represents the rate of exponential decay
of plasma energy when power sources are shut off. In
the definition of confinement time, the loss mechanism is
arbitrary, but in practice losses are dominated by fluctuations.
The fluxes that quantify fluctuation-driven losses depend on quadratic correlations of particular fluctuating
fields. For example, the flux of electrons is governed by
the continuity equation for the electron number density
ne :
⳵ne
⫹ⵜ• 共 ue n e 兲 ⫽0,
⳵t
(2.4)
where ue is the electron flow, specified either in a fluid
description by a continuum equation for the flow, or in
the kinetic description by the velocity moment of the
single-particle probability distribution in the phase space
of electron position and velocity. Toroidal and poloidal
flows effectively lie on nested tori and therefore lead to
no loss, provided the torus of interest intersects no material surface. Net losses are consequently governed by a
radial derivative of the product of density and radial
flow. Separating the density into ensemble-averaged and
fluctuating components 具 n e 典 and ñ e , and considering
equilibria in which the average radial flow is zero,
具 u re 典 ⫽0, we find that the average density is governed by
⳵ 具 n e典 ⳵ ⌫ e
⫹
⫽0,
⳵t
⳵r
(2.5)
P. W. Terry: Suppression of turbulence by sheared flow
where the fluctuation-induced particle flux ⌫ e is
⌫ e ⫽ 具 ũ re ñ e 典 .
(2.6)
In a turbulent flow, the advection of random density
perturbations by the random motions of turbulent flow
must be correlated; otherwise blobs of density are
moved in all directions with equal probability. Such correlations naturally occur when density perturbations are
produced by the random advection of a gradient of the
average density 具 n e 典 . If the density perturbations are
proportional to the gradient, substitution of the proportionality relationship into the flux yields an expression of
Fick’s law (i.e., ⌫ e ⫽⫺D e d 具 n e 典 /dr, where D e is the electron diffusion coefficient).
This review deals with fluctuating flows produced by
E⫻B motion: in a plasma with magnetic and electric
fields, individual particles (Krall and Trivelpiece, 1973)
undergo circular rotation about magnetic-field lines and
drift across the field with a velocity
uE ⫽
共 E⫻B兲
.
B2
(2.7)
The E⫻B drift is independent of charge and identical
for all charge species. It therefore represents a fundamental plasma flow. Under fairly general conditions the
E⫻B flow is the sole advecting flow for fluctuations of
density, temperature, and flow (Kim et al., 1991). In a
tokamak, the mean magnetic field B 0 is primarily toroidal, and the electric field is approximately an electro˜ . The radial component
static fluctuation, i.e., E⫽⫺ⵜ ␾
of the fluctuating E⫻B drift is therefore ũ Er
˜
⫽⫺B ⫺1
0 ⵜ ␪ ␾ , where ⵜ ␪ is the derivative in the poloidal
direction. If a Fourier transform is introduced for the
coordinates in the toroidal and poloidal directions, the
real part of the flux becomes
Re ⌫ e ⫽⫺B ⫺1
0
兺k k ␪ Im具 ␾˜ ⫺kñ k典 ,
(2.8)
˜ ⫺k and ñ k are Fourier amplitudes, k is the wave
where ␾
vector composed of poloidal k ␪ and toroidal k ␾ components, and the ensemble average is an average over toroidal and poloidal angles. In Eq. (2.8), the turbulent advection of density is represented as a sum over scales,
k⫽l ⫺1 , with each component describing the advection
of a blob of scale k ⫺1 by an eddy of the same scale. The
nonzero correlation required for a net flux here imposes
a constraint on the phase angle between the two complex Fourier amplitudes. The flux is maximal when the
relative phase is ␲ /2. Physically, the maximal case can be
visualized as a blob being carried radially across the
eddy diameter in exactly the eddy lifetime, whereupon
the correlation has decayed and an increment of transport is complete. Nonlinear interactions create a new
correlation and the process continues. There is also an
ion flux, given by an identical relation, with ñ representing the ion density.
A similar procedure shows that the heat flux depends
˜ ⫺k and p̃ k ,
on the product of ␾
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
Re Q ␣ ⫽⫺B ⫺1
0
兺k k ␪ Im具 ␾˜ ⫺kp̃ ␣ k典 ,
115
(2.9)
where Q ␣ and p̃ ␣ k are the heat flux and Fourier pressure
amplitude for charge species ␣. The heat flux, Eq. (2.9),
contains both a conductive component, arising from the
temperature contribution to the pressure, and a convective component, arising from the density contribution.
These fluxes are classified as electrostatic because the
electric field is specified solely by a scalar potential gradient. Magnetic fluctuations also produce transport. The
transport is due to the motion of particles along magnetic fields having a radial component due to an instability or turbulence (Callen, 1977; Rechester and Rosenbluth, 1978; Terry et al., 1996). Magnetic-fluctuationinduced fluxes of particles and heat involve correlations
between the radial magnetic-field component and either
the field-aligned current or the field-aligned heat flux
(Prager, 1990).
Just as advection in the density continuity and energy
equations leads to particle and heat transport through
fluxes constructed from appropriate quadratic correlations, advection in the momentum equation leads to
transport of momentum through a momentum flux with
its corresponding quadratic correlation. Turbulent momentum transport was first studied for neutral fluids described by the Navier-Stokes equation. The ith component of the mean flow 具 u i 典 satisfies
d 具 u i典 1 ⳵
1 ⳵
⫽
⌰ ij ⫽
共 ⫺p ␦ ij ⫹2 ␮ S ij ⫺ ␳ 具 ũ i ũ j 典 兲 ,
dt
␳ ⳵xj
␳ ⳵xj
(2.10)
where d/dt⫽ ⳵ / ⳵ t⫹ 具 u j 典 ⳵ / ⳵ x j is the advective derivative
of the mean flow; u i ⫽ 具 u i 典 ⫹ũ i is the incompressible
flow, ⌰ ij is the total mean momentum flux or total mean
flux; S ij ⫽ 21 ( ⳵ 具 u i 典 / ⳵ x j ⫹ ⳵ 具 u j 典 / ⳵ x i ) is the mean rate of
strain; and
␶ ij ⫽ ␳ 具 ũ i ũ j 典
(2.11)
is the Reynolds stress. Equation (2.10) is known as the
Reynolds momentum equation (Tennekes and Lumley,
1972). According to this equation, the mean momentum
may change because of a gradient in the mean pressure,
a gradient in the mean flow leading to viscous losses, or
the mean transport (advection) of fluctuating momentum, as described by the Reynolds stress. A similar
equation, Eq. (4.4), describes the transport of mean momentum in a plasma.
In the absence of dissipation or external driving, total
momentum is conserved. Therefore the Reynolds stress
describes the exchange of momentum between the mean
flow and the turbulence. Because the turbulent velocity
fluctuations have zero mean, 具 ũ i 典 ⫽0, the net mean momentum exchanged is also zero. Turbulence thus rearranges mean momentum, modifying its profile (spatial
variation), without changing the total (spatially integrated) momentum. Turbulent momentum transport is
characteristic of nonuniform flows, and the Reynolds
stress is therefore nonzero. Like the particle and heat
fluxes of Eqs. (2.8) and (2.9), the Reynolds stress re-
116
P. W. Terry: Suppression of turbulence by sheared flow
quires a correlation between two fluctuations, in this
case, two flow components. Mean flow gradients tilt turbulent eddies, thereby making the Reynolds stress nonzero, just as mean density and temperature gradients
distort turbulence to make the correlations of the particle and heat fluxes nonzero.
B. Scaling theory for suppression of turbulence
The first theory to examine the suppression of turbulence by stable flow shear sought an explanation for robust transport barriers observed in fusion experiments.
The theory, which assumes a given shear flow, identifies
a universal feature of advection of turbulence by stable
shear flow. This feature is more general than the experiments and experimental conditions that motivated the
theory, applying to neutral fluids as well as plasmas. By
the same token, suppression of turbulence by flow shear
does not address every aspect of observed barriers,
which also involve the creation of the flow in a complex,
turbulent fluid and in complicated geometries. These
more specialized issues are examined in later sections.
Of two early theories, one focused on the role of flow
velocity (Shaing, Crume, and Houlberg, 1990); the other
(Biglari, Diamond, and Terry, 1990) focused on the role
of flow shear. The latter followed up on earlier work
(Chiueh et al., 1986) that had addressed the behavior of
turbulence in the localized flow shear layer at the edge
of a particularly well-diagnosed device (Ritz et al., 1984).
Subsequent measurements on that device (Ritz et al.,
1990) established many of the qualitative features of the
flow shear theory.
1. Dimensional analysis
The physics of the flow shear theory is most transparent as an exercise in dimensional scaling analysis (DSA;
Terry, Newman, and Mattor, 1992; Ware et al., 1999).
The effect of flow shear on the inertial interactions of
turbulent eddies and advected scalars is described in the
advective derivative, which accounts for eddy decorrelation under advection by random eddy motion and a
mean flow with shear. A scalar quantity ␰, advected in a
two-dimensional (2D) incompressible turbulent flow
with a sheared mean component, is governed by the following equation:
⳵␰
⳵␰
⳵␰
⳵␰
⫹ū 共 y 兲 ⫹ũ ⫹ ṽ ⫽ ␴ ␰ ,
⳵t
⳵x
⳵x
⳵y
(2.12)
where ū is a mean flow in the x direction and ũ and ṽ
are fluctuating flows in the x and y directions. The mean
flow has shear in the y direction, which hereafter is referred to as the shear-wise direction. The flow is assumed stable. At the present level of specificity, ␰ could
itself be a component of the turbulent flow, the magnitude of turbulent vorticity, or a scalar such as density or
temperature. Terms other than the advective derivative
are lumped with any source term into the right-hand
side of Eq. (2.12). Turbulence in plasmas is typically two
dimensional. Rapid streaming of the highly mobile elecRev. Mod. Phys., Vol. 72, No. 1, January 2000
trons along the direction of the magnetic field (here, the
z direction) smoothes fluctuations along the field and
thus restricts significant variations to the xy plane.
Large-scale turbulence in the atmosphere is also two dimensional due to planetary rotation (Pedlosky, 1979).
As an incompressible flow in 2D, the flow can be written
in terms of a stream function ␾ with ũ⫽⫺ ⳵ ␾ / ⳵ y and ṽ
⫽ ⳵ ␾ / ⳵ x.
In a mean flow with shear, the inertial dynamics have
two time scales. One is the eddy turnover rate [Eq.
(2.2)], ␶ ⫺1
e ⬇ũ/ ␦ x⬇ ṽ / ␦ y⬇ ␾ / ␦ x ␦ y, which describes the
rate at which eddies decay. The second time scale ␶ s
arises from the sheared mean flow and opens the possibility that the turbulent dynamics are altered from the
simple picture of Sec. II. A. Consider an eddy whose
shear-wise extent is ␦ y. It undergoes a differential
stretching along the flow by an amount ␦ x in a time
␶ s⫽
␦x
,
␦ yū y
(2.13)
where ū y is the derivative of ū with respect to y, and
␦ yū y is therefore the difference of mean flow speeds
across the eddy. When the stretching length ␦ x along the
flow is equal to the coherence length, or distance to the
next eddy of comparable size, the eddy loses coherence
due to the random advection by other eddies. With ␦ x
specified as the coherence length, ␶ s⫺1 is the shear strain
rate. When the shear strain rate is smaller than the eddy
turnover rate ␶ ⫺1
e , eddies are only slightly distorted by
the flow shear in an eddy turnover time. In this limit
there is essentially no effect on the turbulent dynamics,
apart from a small geometrical distortion of the eddy
flow pattern.
In the opposite limit when ␶ s ⬍ ␶ e , the differential flow
stretches an eddy to its coherence length in a fraction of
an eddy turnover time, making this shortened time the
new eddy lifetime. In this time, fluid parcels within the
eddy have moved across the flow only a fraction of the
cross-flow eddy size ␦ y. Since the eddy has now lost
coherence, this new shear-wise scale,
␦ y s ⫽ ␶ s ṽ ⫽
␶ s␾
,
␦x
(2.14)
becomes the shear-wise coherence length or eddy scale.
Substituting for ␶ s from Eq. (2.13), with the new eddy
scale ␦ y s replacing ␦ y, we find that this eddy scale is
␦ y s⫽
冋 册
␾
ū y
1/2
.
(2.15)
In terms of this new shear-wise eddy scale ␦ y s , the new
eddy turnover time ␶ (s)
e ⫽ ␦ y s / ṽ s is equal to the shear
time, defining a shortened coherence time ␶ (s)
c :
␶ 共cs 兲 ⬅ ␶ 共es 兲 ⫽
␦ys
ṽ s
⫽
␦x
⫽ ␶ 共ss 兲 .
ū y ␦ y s
(2.16)
Therefore, in a strongly sheared flow, turbulence adjusts
itself via the accelerated decorrelation to shorten the
shear-wise correlation length, bringing the eddy turn-
P. W. Terry: Suppression of turbulence by sheared flow
over time into parity with the shear straining time. The
threshold criterion for this effect defines the strongshear limit.
␧ s⬅
␶s
ũ
⫽
⬍1,
␶ e ␦ yū y
(2.17)
where ␶ e is the eddy turnover time defined with respect
to the original shear-wise scale ␦ y, or equivalently, defined in a region away from the flow shear. Note that in
terms of the ambient time scales ␶ e and ␶ s , the reduced
can be written as the geometric
correlation time ␶ (s)
c
mean,
␶ 共cs 兲 ⫽ ␦ x/ ␦ y s ū y ⫽ ␦ x/ū y
1/2
␾ 1/2⫽ ␶ 1/2
e ␶s .
1/2
(2.18)
When an eddy is introduced into a region with flow
shear, its shear-wise coherence length decreases in the
time ␶ (s)
c . On a longer time scale the eddy velocity decreases because shear increases the advective derivative,
⳵
⳵
⳵
⳵
␦ y s ū y
␾s
1
⫹ū 共 y 兲 ⫹ũ ⫹ ṽ ⬇ 共 s 兲 ⫹
⫹
, (2.19)
⳵t
⳵x
⳵x
⳵y ␶c
␦x
␦ y s␦ x
relative to its value in regions with no shear. The turbulent energy cascade rate, which is governed by the advective derivative, now becomes greater than the energy
input rate ␴ ␰ . (If the latter changes, it also changes on a
slower time scale.) The enhanced rate of energy cascade
(and therefore the enhanced rate of dissipation at the
Kolmogorov scale, as explained in Sec. II. A. 1) decreases the turbulence level, or magnitude of ␰, until a
balance between ␴ ␰ and the left-hand side of Eq. (2.12)
is reestablished. Final steady-state values of the shearwise eddy coherence length, fluctuation amplitude, and
correlation time can be obtained from Eqs. (2.15) and
(2.16) and the balance of the turbulent cascade rate with
the source
␾ s␰ s
⫽ ␴ 共␰s 兲 .
␦ y s␦ x
(2.20)
In these expressions the amplitudes ␾, ṽ , ␰, and source
␴ ␰ must be written as ␾ s , ṽ s , ␰ s , and ␴ ␰(s) , indicating
their self-consistent values in the regions of localized
strong flow shear. As a concrete example, consider the
shear-wise turbulent flow ṽ as the quantity ␰. In terms of
a source ␴ (s)
whose dependence on shear remains unṽ
specified, the eddy coherence length, stream-function
amplitude, and correlation time are
冋 册
冋 册
冋 册
␴ 共ṽs 兲
␦ys
⬇␧ s2/3 共 o 兲
␦yo
␴ ṽ
␴ 共ṽs 兲
␾s
1/3
⬇␧ s
␾o
␴ 共ṽo 兲
␶ 共cs 兲
1/3
共 o 兲 ⬇␧ s
␶c
␴ 共ṽs 兲
␴ 共ṽo 兲
1/3
,
2/3
,
⫺1/3
.
(2.21)
Each quantity has been normalized with respect to its
value determined by ambient turbulence alone, and ␾ o
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
117
and ␴ (o)
are related by the balance of the ambient turṽ
bulence cascade rate and ambient source ␾ o2 / ␦ y o ␦ x 2
⫽ ␴ (o)
ṽ . Further simplification requires specification of
the nature of the driving source and therefore depends
on the details of the turbulence. Because the source can
also depend on the flow shear, the response of each
quantity in Eq. (2.21) to the flow shear can differ. For
example, in a self-similar energy cascade driven by stirring at some scale (Ware et al., 1999), the amount of
energy transferred per unit time in every scale is
constant, implying ␾ 3 / ␦ x ␦ y 3 ⫽const. Under this
(o)
␾ s / ␾ o⫽ ␦ y s / ␦ y o ,
and
␴ (s)
arrangement,
ṽ / ␴ ṽ
2
⫺1
⫽( ␾ s / ␾ o ) ( ␦ y s / ␦ y o ) ⫽ ␾ s / ␾ o ⫽ ␦ y s / ␦ y o . The ratios
of Eq. (2.21) then become ␦ y s / ␦ y o ⫽␧ s , ␾ s / ␾ o ⫽␧ s ,
(o)
and ␶ (s)
c / ␶ c ⫽1. In this case there is no change in the
correlation time because of the manner in which the
source is decreased by the constraint of constant energy
injection rate.
In the above discussions, it has been assumed that ␦ x
is invariant inside and outside the region of flow shear.
For the closed flow patterns characteristic of fusion plasmas this is equivalent to assuming that the number of
eddies along the path traced by the flow remains unchanged as the eddies are distorted by the shear. In such
a case there is a scaled reduction in the cross-shear direction for each scale ␦ x along the flow, i.e., eddies on a
range of scales are affected by the shear as described
above. For eddies whose scales are sufficiently small, the
shear time ceases to be smaller than the eddy turnover
time for ambient turbulence, and Eq. (2.17) is no longer
satisfied. Representing ␦ x with the Fourier wave number k x , there is therefore a spectrum subrange at lower
wave number over which shear suppression occurs selfsimilarly, and a subrange over higher wave numbers
where flow shear has a negligible effect on the turbulent
dynamics. If there are inhomogeneities in the x direction
(arising, for example, from the equilibrium, which in
turn affects the turbulence sources), the scale ␦ x may
itself change in the flow shear. From Eq. (2.14), it is
apparent that if ␦ x increases in a region of flow shear,
the shear strain time becomes larger, weakening the effect of flow by slowing down the shear-enhanced decorrelation. Conversely, if ␦ x decreases in a region of flow
shear, the tendency of flow shear to reduce the crossshear eddy scale and eddy amplitude is enhanced.
2. Relative motion of fluid parcels
The results of dimensional scaling analysis can be recovered from more rigorous analyses that invert the differential operator describing advection. The inverted
operator obviously combines the fundamental time
scales ␶ s and ␶ e . The analysis of this and the next subsection establishes the basic conjecture of DSA: that the
eddy turnover time and shear straining time are brought
into parity by the reduction of the shear-wise eddy scale.
The inversion of the advective operator was originally
done for a two-point description of the fluid motion
(Biglari, Diamond, and Terry, 1990; Shaing, Crume, and
Houlberg, 1990). In this case the inversion yields the
118
P. W. Terry: Suppression of turbulence by sheared flow
relative trajectory of two fluid parcels in the turbulent
flow. This method determines eddy scales and the eddy
coherence time from the difference in the character between relative trajectories of fluid parcels that are in the
same eddy and whose motion is correlated, and fluid
parcels that are in different, comparable-sized eddies,
and whose motion is therefore uncorrelated.
The evolution equation of the two-point correlation
具␰(1)␰(2)典 is constructed by taking the product of ␰(2)
with Eq. (2.12) written for ␰(1), adding it to the product
of ␰(1) with Eq. (2.12) written for ␰(2), and taking an
ensemble average. If the equation is expressed in terms
of relative (⫺) and center-of-mass (⫹) coordinates defined by (x ⫾ ,y ⫾ )⫽(x 1 ⫾x 2 ,y 1 ⫾y 2 ), its form is given by
冋
册
⳵
⳵
⳵
⳵
⫺
D ⫺ 共 x ⫺ ,y ⫺ 兲
⫹ū y y ⫺
具␰共 1 兲␰共 2 兲典
⳵t
⳵x⫺ ⳵y⫺
⳵y⫺
⫽␨,
(2.22)
where ū y ⫽ ␦ ū/ ␦ y ⫹ , ␨ ⫽ 具 ␰ (1) ␴ ␰ (2)⫹ ␰ (2) ␴ ␰ (1) 典 is the
two-point source, and D ⫺ is a nonuniform eddy diffusivity whose precise form (Terry and Diamond, 1985) is
obtained by application of a systematic procedure
known as a statistical closure. Under the statistical closure, the eddy diffusivity and the eddy decorrelation rate
on which it depends are functions solely of quadratic
correlations. The diffusivity is dimensionally equivalent
to the eddy turnover rate (D ⫺ / ␦ y 2 ⬇ ṽ / ␦ y) in a strongturbulence (high-Reynolds-number) regime. In a twopoint description, the diffusion describes the scattering
apart of fluid parcels under advection by the random
flow. The relative diffusion process is dominated by diffusion in the y direction, because it is the direction of
inhomogeneity. Both the equilibrium flow and the equilibrium profile of 具␰典 (which contributes to the source)
are nonuniform in this direction. The usual difficulty
with eddy diffusivities, i.e., that they destroy invariance
symmetries of the original evolution equation, is averted
in the two-point closure used in Eq. (2.22) (Dupree,
1972; Boutros-Ghali and Dupree, 1981). Specifically, the
inertial range conservation of the energy 具 兩 ␰ 兩 2 典 under
turbulent advection is guaranteed by the variation of
2
/␦x2
D ⫺ with relative separation: D ⫺ ⬃D o (x ⫺
2
2
2
2
2
2
2
⫹y ⫺ / ␦ y ) as x ⫺ →0 and y ⫺ →0. For x ⫺ ⬎ ␦ x and y ⫺
2
⬎ ␦ y , D ⫺ approaches the constant D o asymptotically.
2
2
and y ⫺
go to zero reflects the
The vanishing of D ⫺ as x ⫺
coherent flow pattern within the eddy, i.e., fluid parcels
separated by a fraction of the eddy scales ␦ x and ␦ y
essentially move together. Parcels whose separation is
greater than the eddy scale diffuse apart with a random
walk whose rate is governed by D o . These properties of
the relative motion of parcels emerge from the solution
of Eq. (2.22) when ū y ⫽0. The scales ␦ x and ␦ y are natural coherence lengths for the relative motion. In the
presence of flow shear, coherence lengths for relative
motion change: a new shorter scale emerges as the
shear-wise separation at which fluid parcels move apart
in an uncorrelated fashion.
To compare the results of two-point theory with the
heuristic dimensional scaling analysis predictions of the
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
prior section, it is necessary to formulate the DSA predictions for a statistical closure model, which contains a
diffusivity instead of the advective derivative u•ⵜu. The
predictions of DSA take a slightly different form for
statistical closure theory than for the primitive equation
because different powers of ␦ y appear in the turbulent
decorrelation rate: in closure theory, the turbulent decorrelation rate has the diffusivity D o divided by the
shear-wise scale squared, while in the primitive equation
the decorrelation rate is the velocity ṽ divided by the
shear-wise scale to the first power. In the closure model
the shear straining time is unchanged and is given by
␻ s⫺1 ⫽ ␦ x/ ␦ yū y . (Here, time scales are expressed as
rates, i.e., ␻ s ⫽ ␶ s⫺1 , in order to distinguish the DSA predictions of this and the prior section.) The turbulent
2
decorrelation time is ␻ ⫺1
e ⫽ ␦ y /D o . The contracted
shear-wise scale is obtained from ␻ e ⫽ ␻ s , yielding
␦ y Ds ⫽(D o ␦ x/ū y ) 1/3. With the equality v ⬘ / ␦ y s
⫽ ␾ / ␦ x ␦ y s ⫽D o / ␦ y D 2 , Eq. (2.14) is recovered. The
s
shortened correlation time in regions of flow shear is
␻ 共cs 兲 ⫺1 ⫽
2
␦yD
s
Do
⫽
⫽
共 D o ␦ x/ū y 兲 2/3
Do
冉 冊冉
␦ y 2/3
␦ x 2/3
D o1/3
␦ y 2/3ū 2/3
y
冊
⫽ ␻ ⫺1/3
␻ s⫺2/3 .
e
(2.23)
The corresponding expression from the prior section,
Eq. (2.18), involves the geometric mean. The extra factor of ␦ y in the diffusion term leads here to a stronger
weighting on the shear term.
Returning to the two-point calculation, the trajectories of relative motion are extracted from the Green’s
⬘ ,y ⫺
⬘ ,t ⬘ ) that inverts the spafunction G(x ⫺ ,y ⫺ ,t 兩 x ⫺
tiotemporal operator of Eq. (2.22) (Dupree, 1972; Terry
and Diamond, 1985). Spatial moments of the Green’s
2
function define stochastic trajectories. The 具 y ⫺
典 trajec2
⬘ dx ⫺
⬘ dt ⬘ y ⫺
⬘2
tory, for example, is given by 具 y ⫺ 典 ⫽ 兰 dy ⫺
⬘ ,y ⫺
⬘ ,t ⬘ ). Evolution equations for these
G(x ⫺ ,y ⫺ ,t 兩 x ⫺
moments are derived from the operator and can be
solved for ␻ (s)
c t⬍1 in terms of initial separations y ⫺ and
x ⫺ , yielding
1
3
2
2
2
⫹x ⫺
具x⫺
典 ⬇ 关 2␧ s⫺2/3共 ␦ x/ ␦ y 兲 2 y ⫺
⫺2␧ s⫺1/3共 ␦ x/ ␦ y 兲 x ⫺ y ⫺ 兴 exp关 ␻ 共cs 兲 t 兴 ,
(2.24)
1
3
2
2
2
⫹2 1/3␧ s2/3共 ␦ y/ ␦ x 兲 2 x ⫺
典 ⬇ 关共 1⫹2 1/3•3␧ s2/3兲 y ⫺
具y⫺
⫺2 ⫺1/3•6 共 ␦ x/ ␦ y 兲 y ⫺ x ⫺ 兴 exp关 ␻ 共cs 兲 t 兴 ,
(2.25)
2 1/3
where ␧ s ⫽ ␻ e / ␻ s as before, and ␻ (s)
c ⫽(2 ␻ e ␻ s ) . Apart
1/3
from the factor of 2 this expression is identical with
the results of dimensional scaling analysis.
Fluid parcels become decorrelated when the relative
separation reaches the eddy scales ␦ y and ␦ x. This defines the parcel correlation time ␶ pc , which can be obtained by inverting the trajectory expressions,
119
P. W. Terry: Suppression of turbulence by sheared flow
冋冉 冊
冉 冊册
␶ pc ⬇ ␻ 共cs 兲 ⫺1 ln
y⫺
2
␧ s1/3␦ y
1⫹6•2 1/3 x ⫺
⫹
3
␦x
where the ␰ n, ␥ is the amplitude of the Fourier-Laplace
transformation of ␰,
x ⫺y ⫺
2
⫺ 共 1⫹2 1/3兲 1/3
3
␧s ␦x␦y
2
.
(2.26)
From this it is clear that the shear-wise coherence length
is
␦ y s ⫽␧ s1/3␦ y⫽ 共 D o ␦ x/ū y 兲 1/3
(2.27)
while the coherence time is
␶ pc ⬇ ␻ 共cs 兲 ⫺l ⫽ 共 2 ␻ e ␻ s2 兲 ⫺1/3
2 ⫺1
⫽ 共 D o / ␦ y Ds
兲 ⫽ 共 ␦ y Ds ū y / ␦ x 兲 ⫺1 . (2.28)
The coherence length along the flow remains ␦ x in the
presence of strong flow shear, as postulated in the previous section. The above analysis reproduces the results
of the DSA, to wit, the shear-wise scale contracts, bringing the turbulent correlation time and shear straining
2 1/3
time into parity. From the expression ␻ (s)
c ⫽(2 ␻ e ␻ s ) ,
and from other expressions of the two-point theory, it is
apparent that fluctuation suppression by flow shear is
independent of the sign of the flow shear or the flow.
This is confirmed experimentally by varying the sign in a
single device (Weynants et al., 1991) or by comparing
barriers in two devices having opposite signs (Levinton
et al., 1995; Strait et al., 1995).
3. Asymptotic analysis
⫹
冉
⳵␰ n, ␥
1⳵
rD n
r ⳵r
⳵r
n2
D ␰ ⫽ ␴ ␰ ⫺ ␰ n 共 t⫽0 兲 ,
r 2 n n, ␥
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
冊
(2.29)
冕
⬁
0
dt exp共 ⫺ ␥ t 兲
冕
2␲
0
d ␽ exp共 in ␽ 兲 ␰ 共 r, ␽ ,t 兲 .
Here ␰ n (t⫽0) is the Fourier amplitude at the initial
time, (ū/r) r ⫽ ⳵ (ū/r)/ ⳵ r 兩 r o is the shear of differential rotation near some position r o , ␥ n ⫽ ␥ ⫹inū/r 兩 r o is the
Laplace frequency Doppler shifted by the mean vortical
flow at r o , and D n is the turbulent diffusivity. As before,
D n is dimensionally consistent with turbulent advection
(D n ⳵ 2 / ⳵ r 2 ⬇ v ⳵ / ⳵ r). Note that in this calculation there is
diffusion in both the radial and the azimuthal directions.
The quantity n/r o is the azimuthal wave number. Its inverse is the azimuthal eddy size, with n being the number of same-sized eddies that can be placed adjacent to
one another in the circle of radius r o .
Equation (2.29) is formally inverted using a Green’s
function
␰ n, ␥ ⫽ 兰 r ⬘ dr ⬘ G n (r 兩 r ⬘ )D n⫺1 关 ␴ ␰ ⫺ ␰ n (t⫽0) 兴 ,
where G n (r 兩 r ⬘ ) is the impulse response of the left-hand
operator of Eq. (2.29) and with the source describes the
spatiotemporal structure of turbulence in the presence
of the differential rotation of the mean flow. To determine this structure, it is sufficient to examine the properties of the eigenmodes ␺ n, ␥ of the Sturm-Liouville
problem corresponding to the homogeneous equation
1 ⳵ ␺ n, ␥ ␥ n
⳵ 2 ␺ n, ␥
⫹
␺
⫺
⳵ r̂ 2
r̂⫹r o ⳵ r̂
D n n, ␥
⫹
The advective derivative with strong flow shear, under
a statistical closure, has also been inverted using
asymptotic analysis to construct the leading-order contribution to the Green’s function in the asymptotic limit
␧ s Ⰶ1 (Terry, Newman, and Mattor, 1992). Asymptotic
analysis allows determination of the unique combination
of terms that dominates the advective operator in the
asymptotic limit and therefore controls the spatiotemporal structure of the turbulence in the presence of the
inhomogeneity of the background shear flow. Terry
et al. (1992) consider an incompressible vortical 2D
mean flow with symmetry in the azimuthal direction, defined with reference to a polar coordinate system whose
origin is the center of the vortex. Adopting this coordinate system, we specify (r,␽) as the radius and angle,
and ( v ,u) as the flow velocities in the radial and azimuthal directions. The mean flow ū(r) is stable and
purely azimuthal, with radial shear. For simplicity, the
shear is treated as linear in the region of interest.
As in the previous section, a statistical closure is applied to the advective derivative. The quantity ␰ satisfies
关 ␥ n ⫺i 共 r⫺r o 兲 n 共 ū/r 兲 r 兴 ␰ n, ␥ ⫺
␰ n, ␥ ⫽
in
n2
␺ ⫽0,
共 ū/r 兲 r r̂ ␺ n, ␥ ⫺
Dn
共 r̂⫹r o 兲 2 n, ␥
(2.30)
where r̂⫽r⫺r o . The limit of arbitrarily strong flow
shear is particularly useful. In this limit, asymptotic
analysis (Bender and Orszag, 1978) reveals the dominant balance between terms of Eq. (2.30) responsible for
the structure of the advected field. The asymptotic limit
of strong shear is parametrized by ␧ s⫺1 ⫽n(ū/r) r r o3 /D n
→⬁, i.e., by the limit in which the shear straining time
becomes arbitrarily short relative to the turbulent diffusion time at the scale r o of the mean flow.
In the strong-shear limit there are two possible balances among the terms of Eq. (2.30). For stationary turbulence ( ␥ →0), Eq. (2.30) becomes singular in the reference frame of the flow at r o as ␧ s⫺1 →⬁. Here
singularity refers to the fact that the highest derivative
of Eq. (2.30) drops out of the equations unless a singular
layer develops allowing ␺ rr to become as large as
n ␺ r̂(ū/r) r D n⫺1 . Other potential balances lead to inconsistencies and therefore do not dominate. Note that this
balance implies a shrinkage of the radial (shear-wise)
scale to make ␺ rr of the same order as the shear term. If
advection in the sheared mean flow has started as an
initial-value problem, this balance is established only after an eddy turnover time. Prior to that time the advectant responds transiently to the flow shear and
␥ n ␺ /D n balances the shear term. This second balance
represents the initial response of the advectant during
120
P. W. Terry: Suppression of turbulence by sheared flow
the first eddy turnover time, or for situations in which
the advectant is carried through a localized zone of
strong flow shear and out before an eddy turnover time.
The latter does not apply in the present case of a vortical
mean flow that closes on itself, but does apply to other
types of flows, for example, the flow inside a nozzle. For
this short time response the diffusivity can be ignored,
i.e., the problem can be linearized. This is the domain of
rapid-distortion theory (RDT; Hunt, Carruthers, and
Fung, 1991). For the longer-time response, linearization
is not possible and the balance of the diffusive term and
shear straining term must be considered. Because the
turbulence is also distorted under this balance, analysis
in the longer time limit can be called balanced-distortion
theory.
The solution of Eq. (2.30) in the limit of balanced
distortion is obtained systematically using a WKB expansion and a formal ordering in terms of the large parameter ␧ s⫺1 ⫽n(ū/r) r r o3 /D n ⫽ ␻ s / ␻ e . In order to determine the time scale over which the dominant asymptotic
balance is established we also take ␥ n as order ␧ s⫺1 . The
WKB expansion is ␺ n, ␥ ⫽exp关␦⫺1⌺m␦mSm(r̂)兴, where S m
is the WKB eikonal of order m in the expansion. The
dominant balance occurs in a singular layer of width ␦
⫽␧ s . Substituting this form into Eq. (2.30) and solving
for the eikonals to second order, we find the eigenfunctions to be Airy functions modified by the cylindrical
geometry,
␺ n, ␥ ⬃ 共 ⫾1 兲 ⫺1/2共 r̂⫹r o 兲 ⫺1/2
冋 冉
冉
␥ n ir̂ 共 ū/r 兲 r
⫺
Dn
dn
2 ⫺in 共 ū/r 兲 r r
⫻exp ⫾
3
Dn
冊冉
1/2
共 ␧ s⫺1 →⬁ 兲 .
冊
⫺1/4
i␥n
r̂⫹
共 ū/r 兲 r
冊册
3/2
,
(2.31)
The Airy function behavior is a direct reflection of the
essential balance between the diffusion ␺ rr and the linear profile of differential rotation assumed in this model.
From the argument of the exponential in Eq. (2.31) the
shear-wise variation occurs on a spatial scale
␦ r⫽
冋
Dn
n 共 ū/r 兲 r
册
1/3
.
(2.32)
Note that this is equivalent to the shear-wise coherence
length [Eq. (2.27)] of the previous section. Equation
(2.31) also indicates that the response is maximum when
兩 ␥ n 兩 ⬇rn(ū/r) r ⬇ ␦ rn(ū/r) r ⫽ ␻ s2/3␻ 1/3
in accordance
e ,
with Eq. (2.28).
The eigenfunction ␺ describes the response of the advectant ␰ in the sheared turbulence subject to the constraints of the formal asymptotic ordering analysis. The
balances and scalings of dimensional scaling analysis are
recovered here from the unique dominant asymptotic
balance in the limit of large shear. This analysis has also
illustrated the distinction between rapid and balanceddistortion theories, a topic to which we return in Sec. III.
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
C. Role of flow shear in linear instabilities
The suppression of turbulence by stable flow shear
appears to be robust and quasiuniversal in magnetized
plasmas, a property stemming from the central role of
advection in 2D turbulence. It is tempting to examine
linear instabilities using heuristic arguments like those of
the section on dimensional scaling analysis. The term
linear instability refers to a collective plasma mode
whose amplitude e folds at the weak intensities under
which linearization of the governing equations is permissible. Although linear instabilities are widely investigated, there is not always a simple connection between
the properties of turbulence and those of an underlying
instability. The spatial and temporal scales of a linear
instability often change once growth is arrested at finite
amplitude by nonlinear effects and turbulence (for example, see Garcia et al., 1985). Notwithstanding these
caveats, heuristic treatments of linear stability have been
formulated, yielding estimates for the level of flow shear
required to modify instabilities (Hassam, 1991; Waltz,
Kerbel, and Milovich, 1994).
Flow shear enters the advective derivative as a shear
strain rate. The simplest hypothesis for linear stabilization is that the shear strain rate exceeds the inverse time
scale of the instability,
␶ s⫺1 ⬎ 兩 ␻ 兩 ,
(2.33)
where ␻ is the larger of the real mode frequency or the
growth rate (Hassam, 1991). In a variant of this criterion, 兩␻兩 is the linear growth rate, calculated assuming no
shear flow, regardless of the magnitude of the real frequency (Waltz, Kerbel, and Milovich, 1994). These criteria suppose that a large shear strain rate deforms the
advective pattern of the normal mode faster than the
rate at which free energy is released into the instability,
thus inhibiting its release. A simple test of this hypothesis is the Rayleigh-Taylor mode, an instability of a
stratified atmosphere. This mode develops for unstable
stratification, i.e., for higher-density fluid above (higher
gravitational potential) lower-density fluid. The growth
rate ␥ is given by the dimensional combination of the
gravitational acceleration and density-gradient scale
length L n ⫽n(dn/dz) ⫺1 : ␥ ⫽(g/L n ) 1/2. Here, n is the
density. A flow U transverse to gravity with linear variation of flow speed in the vertical direction stabilizes perturbations whose wave vector is in the direction of the
flow, provided the Richardson number, J⫽⫺(g/L n )
⫻(dU/dz) ⫺2 , is greater than ⫺2 (Kuo, 1963; Brown,
1980). For the flow U the shear strain rate is ␶ s⫺1
⫽k y ␦ zdU/dz, where k y is the wave number of the perturbation and ␦ z is the scale of the perturbation in the
direction of inhomogeneity. For this instability, ␦ z
and the criterion of Eq. (2.33) approximately re⬃k ⫺1
y
produces the Richardson number criterion. Wave numbers transverse to the flow and gravity are not stabilized,
but if there is a magnetic field in this direction, the mode
is completely stabilized (Lehnert, 1966).
Comparison with careful calculations of the growth
rates of several instabilities that are considered impor-
P. W. Terry: Suppression of turbulence by sheared flow
tant in limiting the confinement of fusion plasmas indicates that Eq. (2.33) is not generally valid.2 Strong flow
shear often stabilizes collective modes driven by gradients of pressure or other equilibrium quantities, but
weaker shear can be either stabilizing or destabilizing,
depending on the collective mode. Moreover, the
mechanisms involved are varied. In some cases, flow
shear enhances the extraction of free energy by modifying eigenmode structure. In general, proper stability criteria must account for the effect of flow shear on eigenmode structure. This involves the combined effect of
flow shear and other plasma inhomogeneities, introducing complications that are not included in Eq. (2.33).
This issue is revisited in Sec. V. A. Despite these difficulties, simulation studies show the criterion of Waltz
et al. (1994) to be a reasonable predictor to within a factor of 50% for the quenching of ion-temperaturegradient turbulence by flow shear (Waltz, Dewar, and
Garbet, 1998).
D. Observations of transport barriers in plasmas
The first observations of a transport barrier in plasmas
were made by Wagner et al. (1982), several years before
it was realized that flow shear was present in the barrier.
The appearance of the barrier coincided with a sudden
increase in confinement time, typically by a factor of 2,
and the steepening of the radial variation of temperature and density in a narrow layer 1–2-cm thick at the
edge of the tokamak plasma. Direct measurement showing reduction of turbulence in the region of steepened
gradients was not made until several years later in other
devices. [It is evident in Fig. 2 of Taylor et al. (1989), and
was reported in Burrell et al. (1990).] Barrier formation
was triggered by the application of additional heating
power during a distinct temporal transition. After the
transition, the plasma energy and confinement time rose
to higher values, the density and temperature gradients
steepened in the barrier region, and emissions of hydrogen Lyman alpha (H ␣ ) radiation promptly dropped, indicating a significant decrease in particle losses. The H ␣
emission was easily measured and, because its intensity
was observed to undergo the most precipitous change of
any quantity, it was adopted as the indicator of the transition time. The pre- and post-transition conditions were
called L mode and H mode, for low and high confinement, respectively. Transitions were observed when the
input power exceeded a threshold. The threshold was
found to depend on a variety of plasma conditions, often
in a complicated way. Through experiments in which the
transition was triggered by a heat pulse propagating into
the barrier region from a disturbance in the plasma core
(Wagner et al., 1984), it was established that power or
2
See, for example, Chen and Morrison, 1990; Hassam, 1991;
Staebler and Dominguez, 1991; Sugama and Wakatani, 1991;
Carreras et al., 1992; Hamaguchi and Horton, 1992; Wang,
Diamond, and Rosenbluth, 1992; Waltz et al., 1995.
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
121
FIG. 3. Evolution of plasma properties through the transition
from L mode to H mode. The transition is marked by the
dashed line at t⫽1.28 s. The density n̄ e and plasma energy ␤ p⬜
begin increasing abruptly at the transition, while hydrogen Lyman alpha radiation H ␣ decreases. From ASDEX Team, 1989.
heat flux into the barrier region was a critical parameter
and might therefore be equivalent to a critical temperature in this region.
Figure 3 shows time traces of plasma energy and the
intensity of H ␣ emissions with the transition marked.
Figure 4 displays the radial variation, or profile, of density and temperature before and after the transition. The
region of steepened gradients, about 1 cm in extent, is
clearly evident. The H mode represents a fundamental
change in the plasma, as indicated by changes in the
empirical scaling of confinement time with plasma parameters (ITER Confinement Database and Modelling
Working Group, 1997) and a reduction of radial correlation lengths (Costley et al., 1993). In the L mode, the
local diffusivity is governed by large global (machinesize) scales, whereas in the H mode the diffusivity is
governed by the smaller scales of charged-particle gyromotion around magnetic-field lines (Petty et al., 1995).
The evolution of the plasma energy and the profiles in
the H mode is consistent with a local transport process
in which heat and particle fluxes are proportional to local temperature and density gradients (Fick’s law) and
the barrier is created by a reduction in the diffusivities.
Consider, for example, an equation for radial ion energy
transport,
⳵pi ⳵
⫹ Q i ⫽P̂ i ,
⳵t
⳵r
(2.34)
where the heat flux Q i is related to the pressure gradient
through a diffusivity ␹ i :
Q i ⫽⫺ ␹ i
⳵
p ,
⳵r i
(2.35)
and P̂ i is the external power supplied to the ions. In
steady state the input power and energy loss balance. If
the energy loss (and therefore the diffusivity) diminishes
suddenly at some radial position, the input power will
exceed the loss, and the pressure will increase, thereby
increasing the plasma energy. As the pressure increases
inside the radius where ␹ i has diminished, the gradient
of p i steepens. Eventually the product of the reduced ␹ i
and increasing pressure gradient grows until a balance is
122
P. W. Terry: Suppression of turbulence by sheared flow
FIG. 4. Radial profiles of temperature and density in ohmic,
L-mode, and H-mode discharges. The H-mode discharge
has steeper gradients than the
ohmic and L-mode plasmas,
neither of which have a transport barrier. From ASDEX
Team, 1989.
reestablished. Although the loss rate reaches its original
value in this example, the plasma energy and confinement time have increased. If P̂ i is switched off, the pressure will decay more slowly because ␹ i is smaller.
The creation of this type of transport barrier has been
widely reproduced and studied in a large number of devices. Subsequent observations have established that
sheared flows are spun up at the transition. The earliest
observations showed that the radial profiles of the poloidal velocity of different ion species change dramatically
at the transition (Groebner et al., 1989). Subsequently it
was shown that similar changes occur in the main ion
flow, whose direction need not be the same as the flow
of impurity species (Kim et al., 1994), and in the E⫻B
flow. As discussed in Sec. IV, the E⫻B flow is of primary interest as the flow that suppresses turbulence.
Plasma flows are of secondary interest as contributors to
the E⫻B flow. The E⫻B flow exhibits rapid radial
variation in the barrier region where steepened gradients are present, with variations of 20 km/s over a centimeter. The radial variation of the radial electric field,
which determines the E⫻B flow, is shown in Fig. 5, before and after the transition. For reference, the ion temperature profile is also displayed, demonstrating that the
large gradient forms in the same location as the region
of strong flow shear. Measurement of fluctuation intensities shows a decrease in the level of potential, density,
and magnetic-field fluctuations in the H mode. The decrease commences with the transition and is localized to
the region of strong flow shear and gradient steepening.
Note that if fluctuations dominate the transport losses, a
decrease in fluctuations leads to a decrease in the transport fluxes, as is apparent in Eq. (2.9), for example,
which in turn leads to a decrease in diffusivities and
thermal conductivities.
E. Experimental and numerical tests
Shortly after the development of the scaling theory
for shear suppression, detailed measurements of turbuRev. Mod. Phys., Vol. 72, No. 1, January 2000
lence properties in a plasma flow shear layer were made
on a device known as the Texas Experimental Tokamak
(TEXT; Ritz et al., 1990). This device utilized a material
surface inserted in the plasma edge to keep the plasma
from the walls of the containment vessel. Near this surface there was a region of localized flow shear in which a
poloidal E⫻B flow changed sign from ⫺3⫻105 km/s to
3⫻105 km/s in a radial layer about 1.5 cm in thickness.
This region of strong flow shear occurred in the cold
edge plasma, making it accessible to electrostatic probes.
These probes provided measurements of radial profiles
of the E⫻B flow, the mean density gradient, profiles of
fluctuation amplitudes in the density and electrostatic
potential (the latter relates to the fluctuating flow
through the fluctuating E⫻B drift), correlation lengths,
and the correlation time.
The measurements showed that the mean density gradient steepened noticeably in the region of strong flow
FIG. 5. Radial electric field and ion temperature profile before
(negative times) and after (positive times) the transition from
L mode to H mode. (a) In H mode the radial electric field is
seen to develop large radial variation in a narrow layer. (b)
The ion temperature profile steepens in the region of the large
variation. The reflectometer data indicate the decrease in fluctuation activity within the transport barrier. From Doyle et al.,
1993.
P. W. Terry: Suppression of turbulence by sheared flow
FIG. 6. Turbulent correlation and shear straining times at positions inside and outside a localized shear layer. Outside the
layer the turbulent correlation time ␶ D is smaller than the
shear straining time ␶ sh . Inside the layer both times decrease
and become equal, as predicted by theory [Eq. (2.16)]. From
Ritz et al., 1990.
shear. The magnitude of fluctuations in the electrostatic
potential and density were observed to decrease as
probes were moved from the central plasma region
through the flow shear layer toward the plasma boundary. Both observations are consistent with a flow-shearinduced transport barrier. Of more direct significance to
the theoretical predictions were observations that the
radial (shear-wise) correlation decreased in the shear
layer. Ritz et al. (1990) also saw indications of a decrease
in the flow-wise correlation length. The shear straining
and turbulent correlation times of the experiment, labeled as ␶ sh and ␶ D to distinguish them from their theoretical counterparts ␶ s and ␶ e , were measured inside and
outside the shear layer. Outside, ␶ sh⬎ ␶ D , indicating
weak shear. The shear straining time dropped by an order of magnitude in the shear layer to a value below ␶ D
as measured outside. The turbulent correlation time also
dropped in the shear layer to the same value as the shear
straining time, making ␶ (s)
c ⫽ ␶ s , as predicted by theory.
The comparison of ␶ D and ␶ sh is plotted in Fig. 6. The
time ␶ lab
c in Fig. 6 is the turbulent correlation time measured in the laboratory frame. It represents a lower
bound on the turbulent correlation time in the plasma
frame. In the shear layer the flow goes through zero, and
␶ lab
c is the same as the plasma frame correlation time ␶ D .
Moyer et al. (1995) measured the suppression of turbulent fluctuations in the H-mode flow shear layer of the
Doublet III-D tokamak (DIII-D), examining the scaling
of the density fluctuation amplitude with flow shear
strength and comparing with theory. As a probe was
moved through the shear layer from an interior position
toward the boundary, a decrease in density fluctuations
was registered. The fact that fluctuations were low in the
extreme edge, where the shear became zero, probably
reflects a change in the turbulence source that is coupled
to boundary effects and the steep-gradient transport
barrier region further in. The data were compared with
the density suppression prediction of Biglari, Diamond,
and Terry (1990), where n s /n o ⬃␧ s2/3 , and an interpolation fit (Zhang and Majahan, 1992) between the strongRev. Mod. Phys., Vol. 72, No. 1, January 2000
123
shear scaling of ␧ s2/3 and a weak-shear scaling of ␧ s2 derived by Shaing, Crume, and Houlberg (1990) for the
regime ␧ s ⬎1. In the negative-shear region on the inner
side of the shear layer the predicted strong-shear scalings are in close agreement with the observations for
strong flow shear strengths that vary by a factor of 5. In
the outer, positive-shear part of the shear layer the data
are in disagreement with the suppression theory. With
increasing shear, there is little change, or even an increase, in the density fluctuations. Moyer et al. (1995)
speculate that this behavior may be caused by a local
change of the plasma state from L mode to H mode or
by an instability driven by the flow curvature.
Numerous experiments have probed the relationship
between the observed confinement enhancements of H
mode and the theoretical threshold for induced suppression of turbulence ␧ s ⬍1 (Burrell et al., 1992; Matsumoto
et al., 1992; Doyle et al., 1993; Ohdachi et al., 1994;
Tynan et al., 1994). In these experiments the shear strain
rate ␻ s ⫽ ␶ s⫺1 increases significantly as the plasma goes
from L mode to H mode. (The shear strain rate is modified by toroidal geometry, as discussed in Sec. V.) In the
H mode, ␻ s becomes considerably larger than ␻ e , indicating that ␧ s ⬍1. In a novel series of experiments, a
technique referred to as magnetic braking (La Haye
et al., 1993) was used to apply an external torque to the
plasma that slowed down the rotation and decreased the
flow shear. The experiments showed a marked increase
in the fluctuation-driven thermal conductivity in regions
where the flow shear had been decreased by the magnetic braking (Burrell et al. 1995; La Haye et al., 1995).
Because the flow shear was manipulated externally, it
could be concluded that there was a cause-and-effect
relationship between the decreased flow shear and the
increased thermal transport. The conclusion of a causal
connection between the flow shear as agent and the decrease of turbulence and transport as response is more
difficult to demonstrate in L-H transitions. There the
spontaneous transition to a different plasma equilibrium
brings numerous, nearly simultaneous changes affecting
not just the magnitude of flow shear, but the profiles of
mean quantities that enter the turbulence sources and
the transport fluxes. The causality issue was further examined in L-H transitions that show an increase in flow
shear prior to the transition. In these transitions the flow
shear changed before the turbulence and transport,
which in turn changed before the transition and its further modifications of both the equilibrium and the turbulence (Burrell et al., 1995, 1996; Moyer et al., 1995).
The same conclusion was reached in an experiment in
which the transition was induced by biasing the plasma
with an inhomogeneous external electric field that grew
slowly in time (Jachmich et al., 1998). The slowly growing shear of the resultant E⫻B flow led to a steepening
of the density profile that began before the transition.
Biasing experiments can also reverse the sign of the flow
and flow shear. Suppression has been observed in both
cases (Weynants et al., 1991), in accordance with theory.
Suppression of turbulence and turbulent transport by
flow shear is a common feature of numerical studies.
124
P. W. Terry: Suppression of turbulence by sheared flow
FIG. 7. Fluctuations in simulations of iontemperature-gradient turbulence in a torus:
(a) contours of ion density fluctuation with no
rotational shear present, and (b) with rotational shear. Structures have a smaller radial
extent in case (b). From Waltz, Kerbel, and
Milovich, 1994 [Color].
Such studies, by choice of model, are limited to particular types of collective fluctuations. All, however, show
strong suppression at sufficiently large shear. A graphic
illustration is provided by simulations of iontemperature-gradient turbulence in toroidal geometry
(Waltz, Kerbel, and Milovich, 1994). Figure 7 shows a
cross section of a torus for two cases, one with radially
sheared rotation and one without. The structures clearly
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
have a reduced radial extent when sheared rotation is
present. In computations, multiple effects contribute to
the suppression, including the balanced-distortion turbulent decorrelation mentioned in Sec. II. C, the stabilization of linear instabilities discussed in Sec. II. E, and the
disruption of the transport flux cross phases, to be
treated in Sec. V. B. For example, all three processes
appear to contribute to the flow-shear-induced suppres-
125
P. W. Terry: Suppression of turbulence by sheared flow
sion of ion-temperature-gradient-driven turbulence
studied by Hamaguchi and Horton (1992). In most numerical studies, flow shear is not a controlled parameter,
but is determined self-consistently by the momentum
transport dynamics to be discussed in Sec. IV. The Reynolds stress, Sec. IV. B, is a key element in the simulations of Carreras et al. (1991), (1992) Ware et al. (1992),
and Guzdar et al. (1993). The nonsymmetry of transport,
Sec. IV.B.5, is also a feature of the work of Guzdar et al.
(1993). Despite these complications, specific aspects of
the balanced-distortion theory are evident. The suppression threshold criterion, Eq. (2.17), predicts which fluctuations are suppressed by flow shear (Carreras et al.,
1991) and also emerges as the suppression condition
when inhomogeneities in the magnetic field are considered (Carreras et al., 1992). The effect of flow shear on
the shear-wise eddy scales and the absence of an effect
on the flow-wise scales are also observed (Guzdar et al.,
1993). Of current interest in simulations is the observed
role of unsteady zonal flows in lowering the level of turbulent fluctuations (Das et al., 1997; Lin et al., 1998).
Zonal flows are Fourier components of the fluctuating
flow field with poloidal and toroidal wave numbers of
zero. Analysis indicates that zonal flows are collisionally
damped (Diamond et al., 1998; Rosenbluth and Hinton,
1998). Details of the damping are still being investigated.
III. RELATED HYDRODYNAMIC PHENOMENA
AND CONCEPTS
There is a close relationship between hydrodynamics
and plasma dynamics, the basis of which is the E⫻B
drift. As a common velocity for all charge carriers, it
allows plasma to move as a flow continuum. If one considers a plasma in a magnetic field directed along the z
axis, the charge continuity equation under the E⫻B
˜
flow is ⳵␳ q / ⳵ t⫹ⵜ•(uE ␳ q )⫽ ⳵␳ q / ⳵ t⫺B ⫺1
0 ⵜ ␾ ⫻z•ⵜ ␳ q
˜ is the electrostatic
⫽0, where ␳ q is the charge density, ␾
potential, and B o is a constant magnetic field. Substitu˜ ⫽⫺␧ o⫺1 ␳ q , then yields
tion of the Poisson equation, ⵜ 2 ␾
˜
⳵ ⵜ 2␾
1
˜ ⫻z•ⵜⵜ 2 ␾
˜ ⫽0,
⫺
ⵜ␾
⳵t
Bo
(3.1)
an equation that is isomorphic to the vorticity evolution
equation of 2D incompressible Navier-Stokes turbulence with a stream-function representation for the flow,
˜ ⫻z. It is thus possible to identify the elecu⫽⫺B o⫺1 ⵜ ␾
trostatic potential, charge density, and charge continuity
equation with the scalar stream function, vorticity, and
vorticity evolution equation of 2D Navier-Stokes turbulence. This simple picture is modified if the fluctuation
scale exceeds the Debye length, the scale over which
thermal electrons short out potential differences (Krall
and Trivelpiece, 1973). In that case ␳ q ⫽0, but the continuity of ion density n i under the polarization drift (a
finite inertia correction to the E⫻B flow) recovers Eq.
(3.1) with the addition of ⳵ n e / ⳵ t from the neutrality condition n i ⫽n e (Hasegawa and Mima, 1977). The E⫻B
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
flow thus possesses vorticity by virtue of the ion polarization drift. Additional effects, particularly those due to
electron dynamics, complicate Eq. (3.1) but do not supplant the basic dynamics of vorticity advection by the
E⫻B flow (Terry and Horton, 1982; Newman et al.,
1993).
Such similarities lead to the question of why suppression of turbulence by flow shear is not a familiar phenomenon in hydrodynamics. The answer is provided by
three basic conditions for simple observation of the suppression effect: (1) the shear flow should be stable; (2)
the turbulence should remain in the region of shear for
longer than an eddy turnover time; (3) the dynamics
should be 2D. These conditions are generally met in fusion plasmas but, as illustrated in the remainder of this
section, are satisfied in neutral flows only under special
circumstances. Shear flow stability is discussed in Sec.
III. A, where it is shown that unstable shear flows, such
as jets, can be stabilized by a rotation gradient. The condition is mathematically analogous to the condition for
stabilization of shear flow in plasmas by the equilibrium
magnetic-field inhomogeneity, as will be discussed in
Sec. V. A. 1. Section III. B considers rapid-distortion
theory, a theory widely used in hydrodynamics for
strong-shearing regimes ( ␶ s ⬍ ␶ e ), but only valid in the
short time limit when ␶ e exceeds the time that turbulence remains in the region of shearing. Strongly sheared
hydrodynamic flows are frequently well modeled by
rapid-distortion theory, hence they do not satisfy the
second condition. Rapid-distortion theory also illustrates why 2D dynamics are desirable for suppression,
showing that common 3D shear flows lead to vortex
tube stretching, which amplifies components of the vorticity instead of suppressing them. Two-dimensional turbulent flows are unusual in hydrodynamics, except under strong stratification or rotation. Section III. C
introduces geostrophic flow, a hydrodynamic flow that
satisfies the three conditions given above. Further examples are given in Sec. VIII.
A. Stable shear flow
In hydrodynamics shear flows are often unstable, generating turbulence instead of suppressing it. A simple
example is the Kelvin-Helmholtz instability. The instability problem is analytically tractable for a piecewise
continuous mean flow with oppositely directed uniform
flows connected by a region of linear shear: ū(y)⫽U o
for y⬎L, ū(y)⫽⫺U o for y⬍⫺L, and ū(y)⫽U o y/L
for ⫺L⬍y⬍L (Chandrasekhar, 1961). Assuming homogeneity in the z direction, the evolution equation for
vorticity governs the temporal growth of vorticity fluctuations,
冉
冊
˜ ␮
d 2 ū ⳵ ␾
⳵
⳵
˜ ⫻z•ⵜ ⵜ 2 ␾
˜⫺ 2
˜,
⫹ū 共 y 兲 ⫹ⵜ ␾
⫽ ⵜ 4␾
⳵t
⳵x
dy ⳵ x ␳
(3.2)
˜ ⫻z is expressed in
where the fluctuating flow u⫽ⵜ ␾
˜
˜ ⫽ⵜ⫻u is the
terms of the stream function ␾ , and zⵜ 2 ␾
vorticity. The last term on the left-hand side is required
126
P. W. Terry: Suppression of turbulence by sheared flow
for instability, i.e., the flow must have a second derivative, usually referred to as the curvature of the mean
flow. For the piecewise continuous flow this is provided
by the discontinuities in slope at x⫽⫾L. Fluctuations
centered at the vorticity maximum y⫽0 are unstable if
k x Lⱗ1, where k x is the Fourier wave number for the x
direction. The growth rate (neglecting viscosity) is
␥ k ⫽U o k x
冉
1
sinh共 2k x L 兲
1
⫺1⫺ 2 2
k xL
2k x L exp共 2k x L 兲
冊
1/2
.
(3.3)
For this and more general flow profiles, a necessary condition for instability is the existence of a vorticity maximum at the point of inflection of the mean flow (Lin,
1955).
Many flows are unstable, including jets, flow past objects, and wall flows (Poisseuille and Couette flows).
Certain flows, however, are stable. Localized vortices in
2D with azimuthal symmetry and appropriate vorticity
profiles are an example. An isolated 2D vortex strip is
unstable, but if there is a background shear flow with a
uniform shear of sufficient strength opposing the shear
of the isolated vortex strip, the strip is stabilized
(Dritschel, 1989). One realization of the background
flow is the differential rotation of a circular vortex. More
generally, rotation tends to stabilize flow-shear-driven
instabilities (Lesieur, 1997). Geostrophic flows in rotating planetary atmospheres can be stabilized by planetary
rotation. Two types of large-scale instability are thought
to be important in the Earth’s stratosphere (Andrews,
Holton, and Leovy, 1987). Barotropic instability arises
from large horizontal curvature of a zonal (east-west)
flow, ⳵ 2 ū/ ⳵ y 2 , where positive ū is an eastward flow and y
is the northward direction. Baroclinic instability basically arises from vertical curvature ⳵ 2 ū/ ⳵ z 2 , but also involves buoyancy and the vertical density stratification of
the atmosphere. These instabilities require the curvature
forces to exceed the stabilizing effect of rotation. Instability is governed by a quantity S R representing the difference between the rotational and curvature effects:
S R⫽ ␤ ⫺
冉
冊
⳵ 2 ū 1 ⳵
⳵ ū
␳ o⑀
,
2⫺
⳵y
␳o ⳵z
⳵z
(3.4)
where ␤ is the gradient of the projection of the planetary
vorticity on the local vertical, ␳ o is the mass density at a
reference height, ⑀ accounts for stratification and buoyancy effects, and S R is the northward gradient of a quantity known as the potential vorticity. A necessary condition for instability is that S R must change sign in the flow
domain (Andrews, Holton, and Leovy, 1987). The barotropic part of this criterion is derived in Sec. III. C.
Large-scale shear flows in the stratosphere, such as the
equatorial jet (Trepte and Hitchman, 1992) and the Antarctic and Arctic polar vortices (Tuck, 1989; Tuck et al.,
1992), are stable most of the time. Occasionally they
become unstable due to an episodic disturbance called a
wave-breaking event (McIntyre and Palmer, 1984). This
event redistributes the flow curvature, and stability is
reestablished.
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
B. Rapid-distortion theory
Rapid-distortion theory describes the evolution of turbulence that transits through a region of strong mean
flow shear. A typical application of this theory is the
advection of turbulence into and through a region of
strong shear, as occurs for example in a nozzle, or a
constriction or bend in a pipe. Because rapid-distortion
theory linearizes the evolution equations, its validity criterion is that the domain time, or the time during which
turbulent structures are advected through the domain of
sheared flow, be shorter than the time scale of the turbulent interactions (or eddy turnover time) for the largest eddies (Hunt, Carruthers, and Fung, 1991). When
this holds, the nonlinearity has insufficient time to
modify the flow, and the flow retains features of the initial state. Under the linearization, turbulence entering
the region of strong flow shear is taken as an initial state
and mapped under a kinematic ray tracing transformation to the evolving configuration at later times. The
mapping is applied to quantities such as fluctuation intensities and scales, energies, and Reynolds stresses. Because the domain time is the time over which the strong
flow shear interacts with turbulence, it is the shear
straining time for finite-length domains. The criterion
for rapid distortion theory therefore resembles the
threshold condition for shear suppression, Eq. (2.17).
However, plasma flows in fusion devices are circular,
and the domain time becomes much longer than both
the shear straining time and the nonlinear interaction
time. In this situation linearization is not valid.
A lucid discussion of rapid-distortion theory is given
by Townsend (1976). Following Batchelor and Proudman (1954), he treats the mean flow variation as linear
and uses the method of characteristics to transform the
inhomogeneity to a characteristic equation for Fourier
wave numbers. With mean and fluctuating flow components U i and u i , the linearized Navier-Stokes equation
is
1 ⳵p
⳵ui
⳵ui
⳵Ui
⳵ 2u i
⫹u l
⫽⫺
⫹␮
,
⫹U l
⳵t
⳵xl
⳵xl
␳ ⳵xi
⳵ x 2l
(3.5)
where for incompressible flow the pressure and flow are
2
related via 2 ␳ ( ⳵ U l / ⳵ x m )( ⳵ u m / ⳵ x l )⫽⫺ ⳵ 2 p/ ⳵ x m
, enabling Eq. (3.5) to be expressed solely in terms of the
flows. The mean flow variation is linear, U l
⫽x j ( ⳵ U l / ⳵ x j ), where ⳵ U l / ⳵ x j is a constant. Introducing
a Fourier expansion u i ⫽⌺ k a i (k,t)exp(ik•x), one can
transform the inhomogeneity to a derivative of a i with
respect to k j ,
Ul
⳵ui
⳵Ul ⳵ui
⫽x j
⫽⫺
⳵xl
⳵xj ⳵xl
兺k
⳵Ul ⳵ai
k exp共 ik•x兲 .
⳵xj ⳵kj l
(3.6)
The amplitude evolution equation is solved by the
method of characteristics, writing da i /dt⬅ ⳵ a i / ⳵ t
⫹( ⳵ a i / ⳵ k j )(dk j /dt). The characteristic equation for
wave-number evolution is obtained from the coefficient
of ⳵ a i / ⳵ k j in the Fourier-transformed Navier-Stokes
equation, yielding
127
P. W. Terry: Suppression of turbulence by sheared flow
da i
⳵Ui
2k i k l ⳵ U l
⫽⫺ ␮ k 2 a i ⫺
a l⫹
a ,
dt
⳵xl
k2 ⳵xm m
(3.7)
dk j
⳵Ul
k .
⫽⫺
dt
⳵xj l
(3.8)
The amplitude a i evolves from an initial value according
to a i (k,t)⫽a i 关 k0 (k,t),t 0 兴 , where k0 (k,t) is the timeinverted solution of Eq. (3.8), with k0 the initial value at
t 0 and k the value at time t.
Two examples of simple mean flow configurations illustrate how flow shear modifies intensities and scales
for short times. The first is an irrotational flow U 1
⫽ ␣ 1 x 1 , U 2 ⫽ ␣ 2 x 2 , U 3 ⫽ ␣ 3 x 3 , where a l can vary in time
but not in space. This is a 3D flow that illustrates how
vorticity is amplified by vortex tube stretching. Incompressibility constrains ␣ l so that ␣ 1 ⫹ ␣ 2 ⫹ ␣ 3 ⫽0. This
type of flow occurs in ducts with changing cross section.
The clearest indication of the distortion is obtained by
applying the method of characteristics to the vorticity
equation,
⳵␻ i
⳵␻ i
⳵Ui
⫽␻l
,
⫹U l
⳵t
⳵xl
⳵xl
(3.9)
where ␻ ⫽ⵜ⫻u is the vorticity, and viscosity has been
assumed to be negligible. Under the Fourier expansion
␻ i ⫽⌺ k ⍀ i (k,t)exp(ik•x), the characteristic procedure
yields
d⍀ l
⫽ ␣ 1⍀ 1 ,
dt
dk 1
⫽⫺ ␣ 1 k 1 ,
dt
(3.10)
with similar equations for the other two components. It
is evident that along directions in which the flow moves
outward from the origin, the vorticity intensifies and the
wave number decreases. The opposite is true along directions in which the flow moves inward. This is a simple
manifestation of a basic process known as vortex tube
stretching (Tennekes and Lumley, 1972). The outwarddirected flow velocity increases with distance from the
origin and thereby stretches vortex tubes whose axes
align with the flow velocity. The increase of vortex tube
length requires a decrease of cross-sectional area because the vortex volume must remain invariant. The
smaller cross-sectional area requires an increase in vorticity to maintain the invariance of circulation
( 兰 u•dl⫽const for inviscid flow). In a flow with ␣ 1
⫽⫺ ␣ 3 ⫽const, ␣ 2 ⫽0, vorticity in the x 1 direction increases while its cross-sectional area, parametrized by
k 3 , decreases. This means that the wave number k 3 increases. Similarly, ⍀ 3 and k 1 decrease. Note that the
intensification of vorticity is a 3D phenomenon. If the
turbulence is 2D, the vorticity is solely in the x 2 direction. With ␣ 2 ⫽0, the vorticity is unchanged. In this case
only the scales are modified. In 3D flows, the intensification of vorticity aligned with the mean flow is a competing effect to the reduction of vorticity perpendicular
to the flow, making suppression of turbulence, even in
the long-time domain, difficult to detect.
A second example is unidirectional plane shearing,
with U 1 ⫽ ␣ x 3 , U 2 ⫽U 3 ⫽0. This flow can be viewed as
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
the superposition of an irrotational plane straining flow
(i)
U (i)
1 ⫽1/2␣ x 3 , U 3 ⫽1/2␣ x 1 , and a rotation about the x 2
(r)
axis U 1 ⫽1/2␣ x 3 , U (r)
3 ⫽⫺1/2␣ x 1 . The irrotational
flow is like that of the previous example, but the axes of
outward and inward flows are turned through 45°. The
tendency of the irrotational flow to align and intensify
vortex tubes along angles of 45° and 225° with respect to
the x 1 axis is counteracted by the rotational part of the
flow, which turns vortex tubes away from these directions. Plane shear flow is therefore less efficient at transferring energy to turbulence than irrotational flow.
Townsend (1976) examines rapid-distortion theory for
unidirectional plane shearing of turbulence with an eddy
viscosity, such as the diffusivity of statistical closure
theory used in Eq. (2.29). It is tempting to view the presence of the eddy viscosity as an extension of rapiddistortion theory away from its traditional linearization
of the dynamics. However, examination of wave-number
evolution shows that rapid-distortion theory remains linear, even with an eddy viscosity in the model. The wavenumber evolution, specified by
dk 1 dk 2
⫽
⫽0,
dt
dt
dk 3
⫽⫺ ␣ k 1 ,
dt
(3.11)
does not depend on the eddy viscosity and is strictly
reversible. The solutions,
k 1 ⫽k 10 ,
k 2 ⫽k 20 ,
k 3 ⫽k 30⫺ ␣ k 1 共 t⫺t 0 兲 ,
(3.12)
have secular evolution, implying no loss of correlation.
The k 3 solution describes the unbounded stretching of
an eddy along the x 1 axis. If some feature of the turbulence has an initial structure
s 共 k 1 ,k 3 ,t 0 兲 ⫽exp关 ⫺ 共 k 21 ⫹k 23 兲 /2⌬k 2 兴 ,
(3.13)
at later times the envelope (irrespective of amplitude
changes) becomes
s 共 k 1 ,k 3 ,t 兲 ⫽exp„⫺ 关 k 21 ⫹ 关 k 3 ⫹ ␣ k 1 共 t⫺t 0 兲兴 2 兴 /2⌬k 2 ….
(3.14)
It is apparent that the width of the structure 具 k 21 典 in the
k 1 direction of wave-number space shrinks in time as
具 k 21 典 ⬇⌬k 2 / 关 ␣ 2 (t⫺t 0 ) 2 ⫹1 兴 . The stretching with time
along the x 1 axis thus continues without bound. The
eddy viscosity contributes to amplitude decay, but the
extended structure maintains coherence indefinitely,
both temporally and spatially.
The secularity and reversibility of the wave-number
solution, Eq. (3.12), with its direct dependence on initial
state, are striking given the presence of eddy viscosity
and the essentially identical content of Eq. (3.5) and the
models of the prior section, e.g., Eq. (2.29). Returning to
Eq. (2.29), the rapid-distortion wave-number solution
can be obtained by retaining the initial state in the
Laplace transform, taking the turbulent diffusivity to
zero and inverting the spatially inhomogeneous operator. The dominant asymptotic balance in Eq. (2.29) with
D n ⫽0 is
⫺ ␥ n ␺ n, ␥ ⫹in 共 ū/r 兲 r r̂ ␺ n, ␥ ⫽ ␺ n 共 r̂,t⫽0 兲 .
(3.15)
Converting this equation to the Cartesian geometry of
128
P. W. Terry: Suppression of turbulence by sheared flow
the plane shearing example, with (ū/r) 兩 r 0 ⫽0, r̂→x 3 ,
and in(ū/r) r →i ␣ k 1 , this balance is
⫺ ␥␺ k l , ␥ 共 x 3 兲 ⫹i ␣ k 1 x 3 ␺ k 1 , ␥ 共 x 3 兲 ⫽ ␺ k 1 共 x 3 ,t⫽0 兲 .
(3.16)
If the initial state is given by ␺ k 1 (x 3 ,t⫽0)
⫽exp关⫺1/2x 23 ⌬k 2 ⫺k 21 /2⌬k 2 兴 [the equivalent of Eq.
(3.13)], the solution of Eq. (3.16) is
␺ 共 x 1 ,x 3 ,t 兲 ⫽exp兵 ⫺1/2关 x 23 ⫹ 共 x 1 ⫺ ␣ x 3 t 兲 2 兴 ⌬k 2 其 ,
(3.17)
where the inverse Laplace transform has been used.
Equation (3.17) is the Fourier transform of Eq. (3.14),
up to a normalization factor, confirming the statement
made in Sec. II.B.3 that rapid-distortion theory is the
short-time asymptotic balance for the strong-shearing
limit. For longer times, specifically when the elapsed
time from the initial moment exceeds the eddy turnover
time or turbulent correlation time, ␥␺ becomes smaller
than the turbulent diffusivity term neglected in Eq.
(3.15), and it enters the balance that determines spatial
structure.
The validity criterion given for rapid-distortion theory
by Hunt, Carruthers, and Fung (1991) should thus be
modified. They stipulated that ␶ D ⬍ ␶ N , where the domain time ␶ D is the time during which flow is advected
through the domain of strong flow shear, and ␶ N is the
turbulent correlation time of the largest eddies. The
modified rapid-distortion criterion is
␶ S⭐ ␶ D⬍ ␶ N .
(3.18)
The distinction between the domain time and the shear
time, which are normally equal in rapid-distortion
theory, is introduced because the two times become distinct in fusion plasma transport barriers where the
sheared flow is circular. The shear suppression criterion
of balanced-distortion theory is therefore
␶ S⬍ ␶ N⬍ ␶ D .
(3.19)
In this limit turbulent diffusivity enters asymptotic balances and affects both scales and intensities.
It is instructive to examine why the eddy diffusivity
appears only in the amplitude evolution of rapiddistortion theory but never enters the wave-number evolution, while in the Green’s-function techniques of the
previous section, it affects both amplitudes and wave
numbers for sufficiently long times. Eddy diffusivities
are a dissipative representation of the energy lost at a
given scale due to turbulent energy transfer, which in
reality is a conservative process. If a conservative model
of the turbulent transfer process is used, the characteristic method of Batchelor and Proudman (1954) leads to
an eddy diffusivity dependence in the wave-number evolution equation. Conservative closures, such as the
direct-interaction approximation (Kraichnan, 1959),
which are derived in a Fourier space representation,
produce spatially inhomogeneous diffusivities in real
space (Boutros-Ghali and Dupree, 1981; Terry and Diamond, 1985). As described in Sec. II.B.2, the diffusivity
vanishes quadratically as the separation of fluid parcels
goes to zero. Using a method of characteristics to sepaRev. Mod. Phys., Vol. 72, No. 1, January 2000
rate out the evolution of wave numbers, we see that the
quadratic spatial variation of the diffusivity enters the
wave-number evolution as an effective force that drives
the second temporal derivative of the wave number. The
analysis of Sec. II.B establishes that this force increases
the shear-wise wave number (reduces the shear-wise
scale), consistent with dimensional scaling analysis.
Balanced-distortion theory is thus an extension of
rapid-distortion theory into a nonlinear regime. The extension, however, is for the simplest of situations and
does not address many complexities that arise even in
the linearized theory. These complexities include 3D behavior and boundary conditions. In fusion plasmas the
magnetic field generally imposes 2D motion, and boundary conditions are often ignored for small-scale fluctuations because the inhomogeneity of the equilibrium
magnetic field localizes fluctuations away from boundaries. In applications of rapid-distortion theory, there is
also frequently a need to achieve highly accurate flow
modeling, which requires mappings that preserve key
symmetries of Reynolds stresses and other tensoral
quantities (Reynolds and Kassinos, 1995).
C. Geostrophic flows and other examples
Hydrodynamic flows that satisfy the three conditions
required for simple observation of turbulence suppression by flow shear do not usually include common types
of shear flow. One that does is geostrophic flow. Consider the Earth’s stratosphere, where mean geostrophic
flows, such as the equatorial jet and the Arctic and Antarctic polar vortices, run zonally (east-west) and have
shear in the meridional (north-south) direction. Provided that planetary rotation is sufficiently strong relative to the meridional curvature to prevent barotropic
instability [Eq. (3.5)], these mean flows are stable. Because of planetary rotation, large-scale geostrophic turbulence is approximately 2D. As zonal flows, the mean
flows close on themselves, making the domain time
larger than any nonlinear time.
The simplest model for geostrophic turbulence is the
quasigeostrophic ␤-plane model. The ␤ plane is a tangent plane to the planetary surface. Fluid motions are
2D in this plane, provided that their eddy turnover time
is larger than the local planetary rotation time, i.e.,
⍀ sin ␪␶e⬅Ro⫺1Ⰷ1. Here ⍀ is the planetary rotation
rate, ␪ is the latitude, ␶ e is the eddy turnover time, and
Ro is the Rossby number. The total vorticity perpendicular to the ␤ plane, which includes the planetary vorticity component 2⍀sin␪, is approximately conserved
under these conditions, yielding d/dt(2⍀ sin ␪⫺ⵜ2 ⌽)
⫽0. The total flow vorticity ⵜ 2 ⌽ is the curl of the incompressible 2D flow u, which can be expressed in terms of
the stream function ⌽: u⫽ⵜ⌽⫻z, where z is normal to
the ␤ plane. The stream function has mean and turbulent components ⌽⫽⌽ 0 (y)⫹ ␾ (x,y), yielding flow u
⫽ū(y)x⫹ⵜ ␾ ⫻z with a zonal mean component with
meridional shear ū(y)⫽ ⳵ ⌽ 0 / ⳵ y and a turbulent component ⵜ ␾ ⫻z. The ␤-plane coordinates x and y are zonal
and meridional displacements from the tangency point
of the ␤ plane. In terms of the longitude ␸ and latitude ␪
P. W. Terry: Suppression of turbulence by sheared flow
129
of any point on the ␤ plane, x⫽a cos ␪0(␸⫺␸0) and y
⫽a( ␪ ⫺ ␪ 0 ), where ␸ 0 and ␪ 0 are the longitude and latitude of the tangency point and a is the planetary radius.
Making advection explicit in the total derivative 关 d/dt
⫽ ⳵ / ⳵ t⫹ū(y) ⳵ / ⳵ x⫹ⵜ ␾ ⫻z•ⵜ 兴 , one finds the vorticity
conservation expression
冉
冊
⳵
⳵
⳵␾ ⳵ ⳵␾ ⳵
⫹ū 共 y 兲 ⫹
⫺
ⵜ 2␾
⳵t
⳵x ⳵y ⳵x ⳵x ⳵y
⫹␤
⳵ ␾ ⳵ ␾ d 2 ū
⫺
⫽0,
⳵ x ⳵ x dy 2
(3.20)
where the northward gradient of the locally vertical
component of planetary vorticity defines ␤:
d/dy 关 2⍀ sin共 ␪ 0 ⫹y/a 兲兴 ⫽2⍀a ⫺1 cos ␪ 0 ⫹O 共 y/a 兲
⬅ ␤ ⫹O 共 y/a 兲 .
The stabilizing effect of ␤ is evident in its creation of an
effective zero-point shift of the mean flow curvature. If
␤ ⫺d 2 ū/dy 2 does not change sign, there is no point of
inflection. The barotropic instability condition [Eq.
(3.5)] is thus a form of the Lin (1955) criterion. The
quasigeostrophic ␤-plane model is closely related to the
Hasegawa-Mima (1977) model of magnetized plasma
turbulence, a relationship that originates with an isomorphism between the Coriolis force in a rotating neutral fluid and the Lorentz force in a magnetized plasma.
From the respective momentum equations, ␳ (du/dt
⫹2⍀⫻u)⫽⫺ⵜp and ␳ du/dt⫽e ␳ m ⫺1 (u⫻B)⫺ⵜp, the
dynamical equivalence of 2⍀ and em ⫺1 B is evident.
This has two consequences of relevance to present considerations: (1) rotation gradient and magnetic-field inhomogeneity play analogous roles in stabilizing shear
flow (the latter is developed in Sec. V. A. 1), and (2)
rotation and magnetic field enforce 2D dynamics.
The effect on geostrophic ␤-plane turbulence of a
large-scale mean zonal flow with meridional shear has
been considered by Shepherd (1987) and Ware et al.
(1999). Part of Shepherd’s analysis employs rapiddistortion theory and describes shear straining of Rossby
wave packets in the absence of nonlinear interactions.
Wave numbers evolve according to Eq. (3.11), and the
unbounded zonal stretching of eddies eventually converts wave energy into the zonal mean flow. The discussion of nonlinear effects focuses on the tendency of the
turbulence to isotropize eddies. While the shear suppression mechanism is not identified, it appears to be
operating. In a simulation of driven, stationary geostrophic turbulence, the near absence of enstrophy
(mean-squared vorticity) in regions of strongest basic
shear is pointed out by Shepherd (1987). This is apparent in a plot of contours of constant enstrophy (meansquared vorticity), shown in Fig. 8. Where the flow shear
is zero in horizontal bands at y⫽0, ␲, and 2␲, enstrophy
fluctuations are strong, as indicated by closed contours
with three and four contour levels for either positive
(solid line) or negative (dotted line) contours. Where
the shear is maximum in bands at y⫽ ␲ /2 and 3␲/2, the
enstrophy fluctuations are weak, as indicated by the apRev. Mod. Phys., Vol. 72, No. 1, January 2000
FIG. 8. Enstrophy contours in simulations of geostrophic turbulence in a ␤ plane with a large-scale sheared zonal jet. Enstrophy is reduced in the regions of large shear at y⫽ ␲ /2 and
3 ␲ /2. From Shepherd, 1987.
pearance of few structures with closed contours, and,
when such structures do occur, the existence of at most a
single contour level. In simulations of decaying turbulence, plots of turbulent energies as a function of the
meridional coordinate (Fig. 15 of Shepherd, 1987) also
show energy minima where the shear is maximum. The
enstrophy contour plots suggest that the ratio of meridional to zonal scale is reduced in regions of strong shear
relative to regions of no shear for the larger scales. For
smaller scales the fluctuations are isotropic. These tendencies are consistent with a reduction of shear-wise
scale for the larger eddies, as indicated by Eqs. (2.15)
and (2.17). In the simulations of Shepherd the suppression of turbulence is marginal because the shear is not
very strong. In Sec. VII ␤-plane turbulence is revisited
to examine stronger shear and confirm the predictions of
balanced-distortion theory.
The ␤-plane model, Eq. (3.20), also has linear wave
solutions called Rossby waves. Linearization of Eq.
(3.20) for ū(y)⫽0 yields the Rossby wave dispersion
␻ R⫽
⫺k x ␤
共 k 2x ⫹k 2y 兲
,
(3.21)
where ␻ R is the Rossby wave frequency, and k x and k y
are wave numbers in the zonal and meridional directions. When a coherent Rossby wave couples to pressure
fluctuations to become a thermal Rossby wave and produce a heat flux [Eq. (2.9)], flow shear can suppress the
transport (Azouni, Bolton, and Busse, 1986; Or and
Busse, 1987). Flow shear changes the complex phase between the fluctuations of pressure and flow, reducing the
transport. Flow shear also changes the complex phase
between pressure and flow in turbulence, as described in
Sec. V. B.
Another class of turbulent flows, referred to as slowly
changing turbulence, has the opposite limit, ␶ N ⬍ ␶ D ,
and thus involves the nonlinearity (Hunt, Carruthers,
and Fung, 1991). Unconfined turbulent flows, such as
wakes, jets, and shear layers, lack confining boundaries
and nominally belong to this class. Observations for
130
P. W. Terry: Suppression of turbulence by sheared flow
nearly a dozen cases of unconfined turbulent flows establish that ␶ N ⫽ ␶ D , suggesting that the turbulence adjusts in some way to achieve this balance. This adjustment is postulated by Hunt, Carruthers, and Fung
(1991) to involve an increase in the largest turbulence
scales to the size of the inhomogeneity in order to bring
the nonlinear time into balance with the domain time.
Note that the plasma flows of balanced-distortion theory
are unconfined in the sense that the domain time becomes large. However, the domain time and shearing
time, unlike rapid-distortion theory, are distinct. The
large domain time allows the nonlinearity to force adjustments, but the inhomogeneity originates with the
shear straining. The nonlinearity thus adjusts the turbulent flow to bring the nonlinear and shear straining times
into balance. In this sense balanced-distortion theory is a
hybrid of slowly changing and rapidly changing turbulence, created by the decoupling of the shearing time
and domain time.
An example in which shearing and nonlinear times
balance is flow over an undulating surface. Changes in
the drag force caused by surface undulations are shown
by Belcher, Newly, and Hunt (1993) to arise from a
thickening of the turbulent boundary layer leeward of
the undulation. Sufficiently far above the surface, the
flow can be modeled with rapid-distortion theory, but
nonlinear interactions become important near the surface in an inner layer. The boundary layer thickening is
caused by the turbulent Reynolds stress in a region
where it is in balance with the shear straining force, i.e.,
where the turbulent correlation time balances the shear
straining time, as in Eq. (2.16); Hunt, Leibovich, and Richards (1988). The shear-wise spatial variation consistent with this balance differs from Eq. (2.15) only in its
details, which reflect the structure of the boundary layer
and the asymptotic matching analysis used to solve the
problem. The scalings, however, evince the same general
features as those of Eq. (2.15): the layer width is proportional to a positive power of the turbulence strength and
a negative power of flow shear strength. Leeward of the
undulation, the shear strength diminishes relative to the
turbulence strength, and the layer thickens.
IV. GENERATION OF SHEARED E⫻B FLOW
Although a variety of flows appear in plasma continuum descriptions, the E⫻B flow is responsible for the
observed suppression of turbulent fluctuations in plasmas. This statement is rooted in empirical observations
and theoretical calculations that show that the E⫻B
flow is the sole advectant of fluctuations in density, temperature, and flow. For example, the diamagnetic flow
u⫽(B 2 en) ⫺1 ⵜp⫻B does not advect fluctuations because it is canceled by a part of the gyroviscous tensor
(see Kim et al., 1991; Chang and Callen, 1992; and the
references to earlier work cited therein). Given the
unique status of E⫻B flow in plasmas, it is essential to
understand its generation. It involves processes that differ dramatically from those of shear-driven hydrodynamic turbulence. In the turbulent boundary layer of a
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
nonionized fluid, for example, the flow shear is determined by the momentum input and the transversemomentum transport arising from the Reynolds stress.
Flow shear drives turbulence, which greatly increases
the drag relative to that of a laminar boundary layer by
increasing the momentum transport to the wall. This
transport regulates the momentum gradient and therefore feeds back on the turbulence drive. Increases in
free energy cause an increase in fluctuation energy, enabling momentum transport to keep pace with the momentum input.
In plasmas, an increase in free energy can lead to a
decrease in fluctuation energy because the free energy
that normally drives turbulence can drive flow shear instead. The result is a decrease in turbulence and turbulent transport, which leads to a further increase in the
gradients providing the free energy (the free energy itself does not necessarily increase because of the suppressing and stabilizing effect of the flow shear). The
generation of mean E⫻B flow shear involves any of a
number of processes affecting momentum, including external momentum sources from the injection of energetic beams of neutral particles or intense rf waves, the
Reynolds stress, mean electric and magnetic fields
through the Lorentz force, the pressure, and dissipative
processes. Turbulence directly participates in the momentum balance governing the generation of E⫻B flow
shear through the Reynolds stress, but also enters indirectly, because turbulent transport affects the pressure
profile and fields.
A. Radial force balance
The problem of determining the mean E⫻B flow in
most plasmas is essentially the problem of determining
the radial component of the mean electric field. The
mean magnetic field can generally be treated as a known
and fixed parameter, because most of the field is produced by external windings or is part of a relaxed state
with a known and robust magnetic field. The electric
field is specified by a mean radial force balance. The
radial component is the relevant component because for
the plasma to be radially confined, both the magnetic
field and the E⫻B flow must lie on the nested tori of
Fig. 2, i.e., they must be perpendicular to the radial direction.
The fundamental equation governing the electric field
is the Poisson equation, ⵜ•E⫽ ␳ q /␧ 0 . The large polarizability of a plasma and the difficulty of directly measuring charge density in fusion plasmas have led to the development of other methods for inferring the electric
field. The Poisson equation is closely related to the momentum, as illustrated in Eqs. (3.1) and (3.2), because
flow is the primary arbiter of charge density through the
charge continuity equation. Consequently, to determine
the radial electric field, it is often sufficient to specify
and solve the radial force balance. It is the ion radial
force balance that is of interest. Ions dominate the
plasma momentum because of their large mass relative
to electrons. The equilibrium radial force balance for
ions is given by
131
P. W. Terry: Suppression of turbulence by sheared flow
E r⫽
⳵
1
mi ⳵
p i⫹
具 ũ ũ 典 ⫺u ␪ i B ␸ ⫹u ␸ i B ␪ , (4.1)
Z i en i ⳵ r
Z i e ⳵ r ri ri
where Z i is the charge state of the ions, e is the electronic charge, n i is the ion number density, p i is the ion
pressure, m i is the ion mass, and the subscripts ␪ and ␸
indicate the components of the mean poloidal and toroidal ion flow and magnetic field. In Eq. (4.1) all quantities have been averaged over magnetic-flux surfaces,
i.e., the surfaces of Fig. 2. Equation (4.1) is comparable
to the Reynolds momentum equation, Eq. (2.10), except
for the absence of dissipation, which is negligible for the
radial momentum component, and the addition of the
electric and Lorentz forces. Two components of the
Reynolds stress are fluxes of radial momentum on
magnetic-flux surfaces and, for incompressible flow, vanish upon averaging. All terms from the advective derivative of the mean flow vanish because, by design, there is
no mean radial flow in magnetically confined plasmas on
the time scales of dynamical evolution. (Transport produces mean radial displacements on longer time scales,
but the effect enters in higher order.) The diagonal Reynolds stress contribution to the radial force balance appearing in Eq. (4.1) is often neglected. Its ratio with the
first
term
of
the
Lorentz
force
is
k 2␪ k r ␳ 3i (Z i e ␾ /T e ) 2 (T e /T i ) 2 (u Ti /u ␪ i ),
where
u Ti
⫽(T i /m i ) 1/2 is the ion thermal speed and ␳ i
⫽(T i m i ) 1/2/Z i eB 0 is the ion radius of gyration in the
magnetic field. The poloidal and radial wave numbers k ␪
and k r are typically smaller than ␳ ⫺1
by an order of
i
magnitude; the electrostatic fluctuation amplitude
Z i e ␾ /T e typically ranges from a few to ten percent, but
is never larger than 1; the ratio of electron to ion temperature T e /T i is of order unity; and the ratio of the ion
thermal velocity to the poloidal flow ranges from 10 to
100. The neglect of the Reynolds stress in the radial
force balance is justified in most situations, although it
could be important in cases in which fluctuations are of
large amplitude and short wave length, and the poloidal
flow is small. Looking at the ratio of the Reynolds stress
to the two other terms on the right-hand side of Eq.
(4.1) leads to similar conclusions. For the ratio with the
pressure term, the same ratio expression given above
applies with u ␪ i replaced by the ion diamagnetic flow
u i (1⫹L n /L T )⫽T i (Z i eB 0 L n ) ⫺1 (1⫹L n /L T ), where
*
L n and L T are the scale lengths of the ion density and
pressure. For the ratio with the toroidal flow term of the
Lorentz force, u ␪ i is replaced by u ␸ i B ␪ /B ␸ . Reynolds
stress components in the toroidal and poloidal force balances are proportionately much more important because
parallel forces are weaker than the radial forces, which
mediate confinement balances. Parallel force balances
are examined in Sec. IV. B.
Ignoring the Reynolds stress in Eq. (4.1), it is evident
that the radial electric field is governed by the ion pressure force and, in conjunction with the mean magnetic
field through the Lorentz force, by the toroidal and poloidal ion flows. Depending on the circumstance, any of
these forces, either individually or jointly, can play an
important role in generating and sustaining the large raRev. Mod. Phys., Vol. 72, No. 1, January 2000
dial electric fields of transport barriers. Measurements
indicate that both the pressure gradient and the poloidal
ion flow become large in the H mode (Figs. 4 and 5).
From experimental observations the ion pressure gradient is unimportant in the initial stages of the H mode but
subsequently becomes the dominant driver of the radial
electric field as reduced transport fluxes steepen the
pressure profile (Burrell, 1994). In core transport barriers, depending on device, the radial electric field is either negative, consistent with a large pressure gradient,
or because of large toroidal rotation it is positive. Observations indicate that poloidal flow is the dominant
driver of the electric field in the initial stages of the H
mode (Moyer et al., 1995). Poloidal flow is also the
dominant driver of the electric field in externally induced transport barriers. In barriers induced by the injection of rf waves, poloidal flow is directly driven by ion
Bernstein waves (LeBlanc et al., 1995). In barriers induced with biased electrodes, a radial current responding to an induced voltage drives a poloidal flow, which in
turn drives the radial electric field (Cornelis et al., 1994).
The toroidal flow is observed to play an important role
in core transport barriers (Greenfield et al., 1993; Burrell, 1997) and in the reversed-field pinch (Fiksel, 1998).
The radial force balance represents a constitutive
equation for the radial electric field. Other relations
must be introduced to determine the pressure gradient
and ion flows. As described by Eq. (2.35), the pressure
gradient is governed by a balance of an energy source
and the heat flux produced by turbulence. The toroidal
and poloidal ion flows are specified from toroidal and
poloidal momentum balances. Because these flows are
key elements of the transition theories that describe the
generation of the radial electric field, the momentum
balances that govern flows must be considered in detail.
B. Mean ion flows
To describe ion flows let us consider first the momentum equation for particle species ␣,
m ␣n ␣
冉
⳵ u␣
⫹u␣ •ⵜu␣
⳵t
冊
⫽q ␣ n ␣ 共 E⫹u␣ ⫻B兲 ⫺ⵜ•p
I␣ ⫹R␣ ,
(4.2)
where R␣ is the mean momentum transfer between unlike particles due to the friction of collisions and I
p␣ is
the total pressure tensor, with isotropic and anisotropic
J ␣ according to I
J ␣ ⫹p ␣ II (Frep␣ ⫽⌸
pressures p ␣ and ⌸
idberg, 1982). Adding ion and electron momentum
equations to eliminate the electric field E yields
mi
冉
冊
⳵ ui
1
⫹ui •ⵜui ⫺ 共 J⫻B⫺ⵜp 兲
⳵t
ni
⫽⫺
冉
冊
J
⳵ ue
ⵜ•⌸
⫹ue •ⵜue ,
⫺m e
ni
⳵t
(4.3)
where J⫽e(n i ui ⫺n e ue ) is the plasma current, p⫽p i
J ⫽⌸
J i ⫹⌸
J e is the total aniso⫹p e is the total pressure, ⌸
132
P. W. Terry: Suppression of turbulence by sheared flow
tropic pressure tensor, and charge neutrality n⫽n i ⫽n e
has been assumed. Equation (4.3) is essentially the
plasma flow equation of MHD plus an electron inertia
term. Electron inertia effects are generally neglected because of the smallness of the ratio of the electron-to-ion
mass. The three remaining terms, the Reynolds stress,
the Lorentz force, and the pressure force (isotropic and
anisotropic), drive mean ion flows. The Reynolds stress
is the primary arbiter of momentum redistribution in
neutral fluid turbulence. In plasmas it plays a central
role in second-order critical theories of the L-H transition (Sec. VII. B. 3). Because measurements of the Reynolds stress in plasmas have been limited, its contribution to flows in fusion experiments is poorly understood.
When mean flow is driven by injected rf waves, the Reynolds stress of the fluctuating wave field drives the flow.
There is a mean Lorentz force due to mean current
when a radial current is driven in the plasma by a biased
electrode. The resulting mean flow triggers a transition
to the externally driven H mode (Sec. VII. B. 1). The
Lorentz force due to the fluctuating current and magnetic field is important in rf-driven flow and in the transition to the enhanced confinement mode of the
reversed-field pinch (Sec. VI. C). The anisotropic pressure tensor involves the most complicated effects in
mean flow drive. It includes viscous damping (Hirshman
and Sigmar, 1981) of mean flows driven by radial pressure gradients in tokamaks (Hinton and Hazeltine,
1976) and the flow drive associated with differential loss
of charge near plasma boundaries. In the following subsections, flow drive associated with the forces of Eq.
(4.3) is discussed in more detail.
1. Reynolds stress
If the isotropic pressure is constant on a flux surface,
the flux-surface-averaged toroidal and poloidal pressure
forces vanish. (The case of pressure with poloidal variation is considered in Sec. IV. B. 5.) For incompressible
flow, the toroidal and poloidal components of u␣ •ⵜ vanish upon averaging, and there are no radial components
of 具 u␣ 典 , 具 J典 , or 具 B典 to lowest order. The only surviving
components of the advective derivatives are Reynolds
stresses. If there are no external currents (external currents are considered in Sec. VII. B. 1), the resulting form
of the poloidal flow equation (Craddock and Diamond,
1991; Diamond et al., 1994) is
冋
册
⳵ 具 u ␪i典
1
⳵
⫽⫺
具 ũ ri ũ ␪ i 典 ⫺
具 b̃ b̃ 典 ⫺ ␮ ␪ 具 u ␪ i 典 , (4.4)
⳵t
⳵r
m in i␮ 0 r ␪
where ␮ 0 is the magnetic permeability, ␮ ␪ is the poloidal
flow-damping rate, and the fluctuating current has been
expressed in terms of the fluctuating magnetic field using
Ampère’s law. In Eq. (4.4), the Reynolds stress,
␶ r ␪ ⫽ 具 ũ ri ũ ␪ i 典 ⫺
1
具 b̃ b̃ 典 ,
m in i␮ 0 r ␪
(4.5)
is generalized to include the effect of the Lorentz force.
The poloidal flow damping originates with the anisotropic pressure force; in a torus with toroidal symmetry,
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
there is no viscous damping of toroidal flow. The poloidal flow damping, called the neoclassical viscosity or
magnetic pumping, arises from ion-ion collisions and the
asymmetry of poloidal variation in a torus (Shaing and
Hirshman, 1989). The poloidal asymmetry is evident in
the variation with poloidal angle of the length of a toroidal arc subtended by a toroidal angle. The arc length is
larger at the outside of a torus ( ␪ ⫽0) than the inside
( ␪ ⫽ ␲ ). There is no asymmetry in the toroidal angle.
The form and magnitude of the neoclassical viscosity
depends sensitively on kinetic effects, and its representation in a fluid description is an area of current investigation (Morris, Haines, and Hastie, 1996; Rosenbluth
and Hinton 1998). Equation (4.4) indicates that, in a toroidal plasma, the balance of Reynolds stress and neoclassical viscous drag determines the radial transport of
poloidal momentum. Clearly it is important to identify
the circumstances under which the Reynolds stress is
nonzero and therefore produces changes in the mean
momentum.
This question has been extensively studied in the
stratosphere, where the Reynolds stress ␳ 具 ũ i ũ j 典 frequently plays an important role in the structure of largescale mean flows. A particularly useful form of the turbulent momentum flux in the stratosphere is the
Eliassen-Palm flux (1961). It is a form of the Reynolds
stress valid for geostrophic flows. Its meridional component in a ␤ plane is simply F ␪ ⫽⫺ ␳ 0 具 u x u y 典 , where ␳ 0 is
the mean density. The vertical component accounts for
vertical momentum transport from the meridional advection of temperature fluctuations in a rotating atmosphere. The divergence of the Eliassen-Palm flux has a
direct relationship to the properties of large-scale linear
wave disturbances: for linear wave action that is steady,
frictionless, and adiabatic, and for conservative mean
flow, the Eliassen-Palm flux divergence vanishes and
there is no acceleration of the mean flow (Andrews,
Holton, and Leovy, 1987). An example of mean flow
acceleration by wave disturbances occurs at a critical
surface, where a mean flow ū(y) with transverse variation is equal to the phase velocity ␻ /k of a propagating
wave. At such a surface the total derivative of the flow,
␻ ⫺kū, vanishes. This allows transience, dissipation, or
nonlinearity, effects that are generally smaller than ␻
⫺kū away from the critical surface, to become important and violate the nonacceleration conditions of the
Eliassen-Palm theorem. The result is a conversion of
wave momentum into mean flow. The momentum deposition can change ū(y) and alter the position of the critical surface. This process is believed responsible for generation of mean zonal flow from vertically propagating
Kelvin waves and Rossby-gravity waves in models of an
observed periodic zonal flow in the stratosphere known
as the quasibiennial oscillation (Holton and Lindzen,
1972; Plumb, 1977). It also figures in the generation of
the seasonal easterly and westerly flow regimes of the
stratosphere (Andrews, Holton, and Leovy, 1987).
In plasmas a similar set of conditions governs the acceleration of mean poloidal flow by the radial derivative
of the Reynolds stress ⳵ / ⳵ r 具 ũ ri ũ ␪ i 典 . To have net mean
P. W. Terry: Suppression of turbulence by sheared flow
flow, acceleration, radial wave propagation, and radial
asymmetry are required. This is readily apparent for
fluctuating E⫻B flow, in which case ũ ri ⫽B ⫺1
0 ik ␪ ␾ k and
ũ ␪ i ⫽B ⫺1
⳵
␾
/
⳵
r.
With
radial
propagation,
⳵ / ⳵ r→ik r ,
k
0
and the Reynolds stress is real; otherwise, it is purely
imaginary. Radial asymmetry is required for the radial
momentum convergence that drives acceleration in a region of interest. Both conditions have corresponding
constraints in the Eliassen-Palm theorem. The plasma
case has been analyzed to delineate circumstances under
which these conditions are met (Diamond and Kim,
1991). In a plasma, radial asymmetry is assured for
large-scale waves, which sample the variation of equilibrium density and temperature on the largest scale (Hasegawa and Wakatani, 1987; Carreras, Lynch, and Garcia, 1991), for wave structures within a radial correlation
length of the plasma boundary or some other significant
boundary (Diamond and Kim, 1991) and for waves in
regions where there is strong variation of the mean density, temperature, or flow. The full plasma Reynolds
stress ␶ r ␪ has been examined for electromagnetic Alfvén
waves (Craddock and Diamond, 1991). In analogy with
flow acceleration in the atmosphere at the critical surface, the wave number of radially propagating Alfvén
waves goes to zero at the resonant surface defined by
␻ ⫽k 储 v A , where k 储 ⫽k•B is the wave vector along the
equilibrium magnetic field, and v A ⫽B/( ␮ 0 m i n i ) 1/2 is
the Alfvén velocity. In this region there is strong resonant absorption of the wave energy and strong radial
variation or asymmetry. The two components of the
Reynolds stress nearly cancel for electromagnetic waves.
For kinetic shear Alfvén waves an imbalance is provided
by ion inertia in the form of the polarization drift (Craddock and Diamond, 1991).
The radial asymmetry required for finite Reynolds
stress can arise from flow shear through its distortion of
fluctuations (Diamond et al., 1994; Terry et al., 1994) or
from other inhomogeneities such as the pressure. For
example, turbulence in an infinite, homogeneous medium has a zero Reynolds stress because advection
across any surface is equally likely to carry both signs of
momentum. This property is traceable to the lack of a
gradient in mean flow, i.e., the lack of an asymmetry in
some direction. Likewise, the Reynolds stress of global
magnetic turbulence in the reverse-field pinch edge is
measured to be zero under most conditions (Fiksel,
1998), a property that can be attributed to the lack of
radial propagation of simple resistive MHD instabilities
when there is no background shear flow (Diamond and
Kim, 1991). The notion of eddy viscosity follows directly
from the dependence of the Reynolds stress on flow
shear. The eddy viscosity is derived from the Reynolds
stress using Prandtl’s mixing-length hypothesis to model
the turbulent flow as linearly proportional to the flow
shear (for a simple derivation, see Tennekes and Lumley, 1972). Eddy viscosity is an oversimplification of the
dynamics, but it illustrates the property that flow shear
leads to mean flow acceleration by turbulence. This is an
important element of second-order phase transitions, as
discussed in Sec. VII. B. 3. From the above discussion,
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
133
note that the conditions for acceleration under wave action essentially carry over to the turbulent case, in which
the motions need not satisfy any collective resonance or
have wavelike features.
The Reynolds stress is difficult to measure in fusion
plasmas and consequently has figured more prominently
in theory than in the interpretation of experiment. Measurements in stellarators (Matthews et al., 1993; Hidalgo
et al., 1997) indicate that the Reynolds stress in those
plasmas is of comparable magnitude to mean flow accelerations. These measurements provide insufficient detail
to establish a causal relationship. Reynolds stresses are
easily measured in simulations and are found to produce
flows of sufficient magnitude to decrease turbulence levels in accord with the principles of Sec. II (Carreras,
Lynch, and Garcia, 1991). The generation of flows by
the Reynolds stress is particularly noticeable in simulations of ion-temperature-gradient-driven turbulence in
tokamak plasmas (Waltz, Kerbel, and Milovich, 1994).
Reynolds-stress-driven flows play a significant role in
saturating the instability. A like process associated with
an instability of flow parallel to the equilibrium magnetic field is also illustrated in Fig. 16 of Charlton et al.
(1994).
2. External biasing
In external biasing experiments, the Lorentz force
drives flows through the balance in Eq. (4.3) of J⫻B
J , where J is the radial current injected by the
⫽ⵜ•⌸
probe. The flow is proportional to the current when the
anisotropic pressure force is dominated by viscous dissiJ ⬀n i m i ␮ ␪ u ␪ ). The radial current
pation (e.g., B•ⵜ•⌸
flows in response to an applied voltage difference between the vessel wall and a surface inside the plasma
(Sakai, Yasaka, and Itatani, 1993) or an electrode inserted into the plasma (Taylor et al., 1989; for a review
see Weynants and Oost, 1993). When the current from
the electrode to the wall is plotted as a function of the
voltage between these points, a bifurcation of the
plasma equilibrium is evident, manifested as an abrupt
change in the effective radial plasma resistance (Taylor
et al., 1989). The bifurcation occurs at a critical value of
the current. Above the critical current the plasma resistance increases sharply, allowing a larger radial electric
field for a given radial current. The current-voltage response for a biased plasma is shown in Fig. 9, along with
the inferred plasma resistance. The jump to a larger field
yields a jump in the magnitude of the E⫻B flow and
flow shear, with concomitant reductions in fluctuations
and transport (Taylor et al., 1989; Weynants et al., 1991;
Tynan et al., 1992; Askinazi et al., 1993). The bifurcation
produces changes in plasma transport and equilibrium
profiles that are similar to those that occur in the
H-mode transition (Sec. II. B). This behavior is consistent with a simple model of the flow dynamics that balances the external poloidal force of the biasing with a
nonlinear flow-damping process that, as a function of
flow speed, first increases to a maximum (at a flow speed
of order the relevant sound speed) and then strongly
134
P. W. Terry: Suppression of turbulence by sheared flow
FIG. 9. Electrode current vs bias voltage in plasma biasing
experiment. The current has a linear response until the point
labeled bifurcation, after which the current actually decreases
and then rises slowly. The inferred plasma resistance is also
plotted. From Taylor et al., 1989.
decreases (Weynants et al., 1991). This allows for two
markedly different flows for the same forcing or external
current. The neoclassical viscosity has a nonlinear form
with a peak near u ␪ ⫽u Ti B ␪ /B ␸ (Shaing and Crume,
1989). Above this critical speed, ions rarely suffer Coulomb collisions and the dissipation of poloidal momentum becomes weak. It is not known if the Reynolds
stress contributes to the force balance or is overwhelmed by the external poloidal force in biasing experiments. Biasing experiments also have been conducted in a variety of nontokamak devices, including the
stellarator (Isler et al., 1992), reversed-field pinch (Craig
et al., 1997), and tandem mirror (Sakai, Yasaka, and Itatani, 1993). All observe reduced transport, but a bifurcation has not been observed in the reversed-field pinch.
Features of bias-induced transport barriers are discussed
in Sec. VI. A. 2.
3. Injected radio-frequency waves
Earlier in this section it was noted that the fields of
electromagnetic waves can generate flow through the
Reynolds stress, Eq. (4.5), where it is understood that
the fluctuating flow ũ of the Reynolds stress is the E
⫻B flow created by the electric field of the wave. Such
waves can be externally injected into a plasma using an
rf antenna. Fluid analyses of the rf force arising from
kinetic shear Alfvén waves (Craddock and Diamond,
1991) and ion Bernstein waves (Biglari et al., 1991) suggest that it may be feasible to excite flow shear in this
fashion. A recent kinetic treatment of the rf force finds
that ion Bernstein waves are less efficient at driving
flows than originally thought, but that fast magnetosonic
waves can drive significant flows at power levels like
those used to heat plasmas (Berry, Jaeger, and Batchelor, 1999). Using rf waves to drive flows is not a simple
procedure. Reflection, scattering, dispersion, mode conversion, dissipation, and differing frequency-dependent
interactions with electrons and ions all enter the problem.
Evidence that a flow-shear-induced transport barrier
can be produced in a core plasma using injected rf waves
has been reported by LeBlanc et al. (1995). This work,
conducted in the Princeton Beta Experiment Modification (PBX-M), involved the resonant absorption of an
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
ion Bernstein wave (IBW) in the core of a plasma that
was already in H mode. It was observed that in the region of IBW absorption, steep gradients developed in
the electron density and ion and electron temperature
profiles. Inside these gradients, the density and temperature reached higher values than those of comparable
plasmas without the IBW. These features suggest a
transport barrier in the region of wave absorption. The
increase of temperature associated with an IBW, and a
concomitant increase in neutron production rate,
abruptly began some time after the IBW power was applied. This time was labeled as the onset time for the
barrier, which was designated as the CH mode (core H
mode). The toroidal flow and toroidal flow shear also
increased with the IBW. There was no diagnostic for
measuring the poloidal flow. (Toroidal flow contributes
to the radial electric field and E⫻B flow shear, but its
contribution is smaller by a factor of B ␪ /B ␸ than that of
a poloidal flow of the same magnitude.) Energy balance
calculations indicated that the transport of toroidal momentum and ion energy diminished at the H-mode transition with a further decrease beginning at the onset of
the CH phase. There was no change in electron energy
transport with either transition. Values for the ion thermal conductivity, toroidal momentum diffusivity, and
particle diffusivity have since been reported as lower
than collisional diffusivities calculated for a toroidal
plasma (Ono et al., 1997). Other devices have had difficulty producing core barriers with rf flow drive, raising
questions as to the robustness of the method. Experiments conducted on the Tokamak Fusion Test Reactor
(TFTR) indicated that shear flow was driven by an ion
Bernstein wave, but no evidence of a transport barrier
was seen, presumably because the flow strength was insufficient to reach the suppression threshold (LeBlanc
et al., 1999). The desirability of controlling the location
and strength of internal shear layers in plasmas warrants
further work on this technique.
4. Orbit loss
In a toroidal plasma, a radial electric field can build up
due to differential charge loss (Itoh and Itoh, 1988) and
differences in the way ion and electron orbits intercept
material surfaces in the edge region (Shaing and Crume,
1989). The latter is referred to as ion-orbit loss and it
provides the basic flow drive in the two-step models of
the H-mode transition described in Sec. VII. Here the
physics of the ion-orbit loss mechanism is presented
heuristically. Mathematically, the orbit loss drive can be
J in
included as part of the anisotropic pressure force ⵜ•⌸
certain regions of the plasma, e.g., in the vicinity of a
magnetic-field-line separatrix. The separatrix is a flux
surface that separates an outer region of the plasma volume, in which diverted magnetic flux intersects material
surfaces, from an inner region in which it does not. The
separatrix and flux surfaces typical of diverted tokamaks
are depicted in Fig. 10. The separatrix geometry is intentionally introduced into fusion plasmas in order to channel the outflow of hot plasma particles. This is achieved
with a device known as a divertor, which consists of external magnetic coils to produce the field geometry, the
P. W. Terry: Suppression of turbulence by sheared flow
FIG. 10. Cross section of flux surfaces in a diverted tokamak
plasma. A surface known as the separatrix crosses itself at a
null point of the poloidal field. Particles moving along field
lines on the separatrix hit a material surface known as a strike
plate. A projection onto a plane of constant toroidal angle of a
banana orbit about one of the flux surfaces is indicated for
illustration.
strike plates on which field lines terminate, and pumping
ducts for edge pressure control.
The loss of particles near a separatrix relates to the
average motion of charged particles trapped in the magnetic field well along magnetic-field lines in a torus. If
the circular motion of charged particles in planes perpendicular to magnetic-field lines is averaged over, the
resultant average motion of charge follows orbits called
guiding centers. The curvature and gradient of the magnetic field cause the guiding centers to drift off the field
lines (Krall and Trivelpiece, 1977). The projection of the
guiding-center orbit onto a cross section at fixed toroidal
angle traces out a banana shape (depicted in Fig. 9)
whose maximum radial thickness is given by
␳ ␪⫽
2u 储 m j
,
兩e兩B␪
(4.6)
where u 储 is the guiding-center velocity along the field. u 储
scales with the thermal velocity, making the ion banana
width larger than the electron banana width by the
square root of the ion-to-electron mass ratio. The radial
gradient of the steady E⫻B drift created by a mean
electric field significantly reduces the larger ion banana
width (Berk and Galeev, 1967). However, the squeezed
ion radius remains much larger than the electron radius.
A collision between trapped particles on banana orbits
shifts a particle on the outside of a banana to the inner
side of a new banana whose inner side is at the position
of the outer side of the old banana. The collision has
thus moved the particle one banana width in the radial
direction, i.e., the Brownian motion random-walk step
size of trapped particles in a magnetic field is the banana
width. Trapped particles within one banana width of the
separatrix move across it upon colliding, and onto field
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
135
lines that intersect material surfaces. If the collision detraps the particle, it moves out of the mirror and along
the field to the strike plate, where it is lost from the
plasma. Because the electron banana width is essentially
negligible in comparison to the ion banana width, the
region within one ion banana width of the separatrix
suffers a net loss of ions and becomes negatively
charged.
Orbit loss also occurs in nondiverted tokamaks near
material surfaces called limiters. The poloidal limiter is
an annular disk placed at a particular toroidal position.
Its outer edge is fixed to the wall, and its inner edge
protrudes radially into the plasma. Because ion banana
orbits can reach behind the limiter without hitting it, the
region one banana width behind the limiter becomes
positively charged (Hazeltine, 1989; Tendler and
Rozhansky, 1992), producing a positive sheath potential
(Ritz et al., 1984). Orbit loss is only capable of producing
a radial electric field within one banana width of a material surface such as a limiter, or of a divertor separatrix, and is therefore intrinsically an edge effect.
5. Nonsymmetric transport
Transport fluxes that are asymmetric in poloidal angle
have been postulated to produce poloidal rotation. The
original hypothesis (Stringer, 1969) predicted flow acceleration that was too weak to overcome neoclassical flow
damping. It has been argued that the fluctuation-driven
fluxes of present-day tokamaks are of sufficient magnitude to drive robust poloidal rotation in the presence of
neoclassical viscosity (Hassam et al., 1991). There is
some evidence of poloidal asymmetries in experiment,
inferred from poloidal variations of the edge plasma
density (Brower, Peebles, and Luhmann, 1987; LaBombard and Lipschultz, 1987) and the observation of large
flows along the equilibrium magnetic field (Vershkov,
1989). Poloidally asymmetric particle transport, poloidal
variation of the density, and parallel flows are directly
linked in the theory.
Poloidal flow arises as part of a collective plasma
mode that couples to the asymmetry of the particle flux
and is uncovered in an ordering scheme based on the
smallness of the inverse aspect ratio a/R 0 (see Fig. 2).
At lowest order there is a poloidally uniform equilibrium density but no steady poloidal flow. Higher order
yields an exponentially growing normal mode with poloidally nonuniform perturbations in u 储 and n, matching
the variation of the particle flux, and a uniform perturbation in u ␪ . The growth rate is proportional to the 31
power of the asymmetric part of the steady parallel flow
and is thus proportional to the same power of the asymmetric part of the particle flux. To obtain large poloidal
flows it is not sufficient to have a large particle flux; a
large asymmetry is required. This process represents a
collective excitation coupling multiple equations, and it
is more complicated than any single force like the Reynolds stress. It has not been determined from experiment if this collective mode is excited in fusion plasmas.
136
P. W. Terry: Suppression of turbulence by sheared flow
C. Miscellaneous processes
The time derivative of the Poisson equation and the
evolution equations for the average ion and electron
densities [e.g., Eq. (2.5)] lead to an expression that may
be evaluated for the radial electric field if the radial
fluxes of ions and electrons are known (Itoh and Itoh,
1988, 1996). In steady state, the dependence of the fluxes
on the electric field must be specified in order to determine the field. The effects of Sec. IV. B may be expressed as contributions to the net difference between
electron and ion fluxes. Any additional process that contributes to the difference of fluxes can affect the radial
electric field, in principle. One such process is the particle transport induced by a slight variation in magneticfield strength with toroidal angle. This variation, called
magnetic-field ripple, is caused by a corresponding
variation in the external current density that produces
the toroidal field. It occurs, for example, because the
external current is driven in a discrete set of coils surrounding the plasma at different toroidal locations. Another process is charge exchange, i.e., the exchange of an
electron between ions and neutrons, or ions and ions.
These contributions to the flux difference have been included in general expressions (Itoh and Itoh, 1996). The
penetration depth of neutrals has been considered in
connection with the location of the H-mode transport
barrier (Carreras et al., 1998), and ripple losses have
been invoked in a transition model for core transport
barriers (Shaing et al., 1998).
V. SUPPRESSION IN TOROIDAL PLASMAS
The basic notion of shear suppression of fluctuations
was presented in Sec. II in a simple form, using a model
in which the only temporal scales were the shear straining and turbulent decorrelation rates, and the geometry
was rectilinear. In this section complications related to
magnetically confined plasmas are addressed. The most
important of these are the inhomogeneity of the magnetic field and toroidal geometry. These issues allow further examination of several distinct mechanisms by
which shear suppresses turbulence, viz., through its effect on turbulent decorrelation, on linear stability, and
on the cross phase of transport fluxes.
A. Plasma geometry and magnetic topology
In a toroidal plasma the magnetic field is not uniform.
Its magnitude varies both along and transverse to the
field lines. The longitudinal variation traps particles (the
orbit loss mechanism of Sec. IV. B.4 is caused by
trapped particles) while the transverse variation, referred to as magnetic shear, has a strong stabilizing effect on collective instabilities. Hence magnetic shear is
utilized to gain stability. Generally, magnetic shear is the
most significant inhomogeneity affecting the eigenmode
structure of collective instabilities. One component of
the magnetic field typically changes its magnitude more
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
FIG. 11. Magnetic-field lines at two radial locations relative to
a radially extended collective plasma eigenmode whose displacement is represented as a sinusoidal sheet. The field line at
r is perpendicular to the wave vector k, so there is no projection of k on B. At r⫹⌬r magnetic shear has twisted the field
line, producing a nonzero projection of k on B.
than the other, so that shear is seen as a change in fieldline direction or pitch from one flux surface to another
(see Fig. 2).
Magnetic shear is stabilizing because collective motions, which extend radially in order to tap the free energy of gradients, experience enhanced damping from
the variation of field-line pitch with radius. This damping is associated with parallel motion, which is efficiently
dissipated by collisions because of the ease of motion
along the field. To couple to parallel motion, collective
modes must have variation along the field, i.e., there
must be a nonzero wave number k 储 along the field. A
collective mode cannot avoid parallel damping if there is
magnetic shear. The mode can arrange for k 储 to be zero
at one radial location, but it will extend to other radial
locations, where k 储 is nonzero, and there is parallel
damping. This is illustrated in Fig. 11, which shows a
radially extended sinusoidal fluctuation with a symmetry
direction parallel to the magnetic-field line at r 关 B(r)•k
⫽k 储 (r)⫽0 兴 . Because of the twist of the field line, k 储 (r
⫹⌬r)⫽0, i.e., B(r⫹⌬r)•k⫽0.
The parallel wave number is calculated from the fieldaligned gradient,
再
冎
B•ⵜ
1 B␸ ⳵
B␪ ⳵
⬅ⵜ 储 ⫽
⫹
,
兩 B兩
兩 B兩 R ⳵␸
r ⳵␪
(5.1)
where the coordinates R, r, ␪, and ␸ are those of Fig.
2(b). Introducing the Fourier transform in toroidal and
poloidal angles, f(r, ␪ , ␸ )⫽⌺ m,n f m,n (r)exp(in␸⫺im␪),
we find the parallel wave number, obtained from
(2 ␲ ) ⫺2 兰 exp(in␸⫺im␪)ⵜ储f(r,␪,␸)d␪d␸⫽ik储fm,n(r), given
by
k 储⫽
再
冎
1 iB ␸ n iB ␪ m
.
⫺
兩 B兩
R
r
(5.2)
In tokamaks with a/R 0 ⬍1, the variation of B ␸ and R
with r is weak compared to that of B ␪ . At a certain
radius r s , where B ␸ n/R⫽B ␪ (r s )m/r s , k 储 vanishes. The
toroidal surface with r⫽r s is called a rational surface,
because the ratio of the two terms in Eq. (5.2),
q共 rs兲⬅
B ␸r s
m
⫽ ,
B ␪共 r s 兲 R n
(5.3)
137
P. W. Terry: Suppression of turbulence by sheared flow
is a rational number on that surface. The quantity q(r)
is the winding number of magnetic-field lines on any
given flux surface, i.e., the ratio of displacement in toroidal angle to displacement in poloidal angle between
two points on a field line. [The use of polar coordinates
for the toroidal cross section and the assumption a/R 0
⬍1, adopted for simplicity of presentation, make Eq.
(5.3) valid only for large-aspect-ratio, circular crosssection flux surfaces. Generalizations of q exist for the
finite-aspect-ratio, noncircular cross-section toroidal
plasmas in common use.] Away from the rational surface B ␪ can be expanded in a Taylor series. Keeping
only the lowest-order nonvanishing term,
k 储⫽
共 r⫺r s 兲 m 共 r⫺r s 兲
⫽
k␪ ,
Ls rs
Ls
(5.4)
where k ␪ is the poloidal wave number, L s ⫽Rq/ŝ 兩 r s is
the magnetic-shear scale length, and ŝ⫽rq ⫺1 dq/dr.
From Eq. (5.4) the parallel wave number is proportional
to the poloidal wave number and has a radial variation
away from the rational surface that is linear with a scale
length L s . The scale length is of the order of the major
radius R and is large compared to the radial scales of
fluctuations.
In the absence of complicating effects such as flow
shear, fluctuations tend to have maximum amplitude at
the rational surface where the damping of parallel motion is zero. The amplitude decays away from the rational surface on a scale that is set by the tradeoff of enhanced dissipation at finite k 储 and the efficient
extraction of gradient free energy. Instabilities are often
analyzed in the so-called slab reduction of toroidal geometry, where
r⫺r s →x,
m/r⫽k ␪ →k y ,
n/R⫽k ␸ →k z .
(5.5)
Note that x is the direction of inhomogeneity, by the
convention of fusion literature. (Up to this point in the
review, the convention of fluid dynamics and atmospheric literature has been followed, making y the direction of inhomogeneity. Both conventions will be used
hereafter, depending on the context.) The effect of magnetic shear on plasma instabilities is illustrated by the
Kelvin-Helmholtz instability, which is driven by flow
shear, but can be stabilized by magnetic shear.
1. Shear flow stability with magnetic shear
The Kelvin-Helmholtz instability (Sec. III. A) is flowshear driven, drawing free energy by the exchange of
vortex filaments about a vorticity maximum. Recalling
from Eq. (3.1) that vorticity is equivalent to charge density, the exchange of vortex filaments produces electric
fields that drive current and dissipate energy according
to Ohm’s law. In the presence of an equilibrium magnetic field, reduced MHD equations (Strauss, 1976) provide a convenient description of this process. Like
MHD, reduced MHD consists of an equation for Ohm’s
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
law and a vorticity evolution equation that is essentially
the curl of Eq. (4.3) without the small electron inertia
terms. However, the mean magnetic field leads to dynamics that are nearly 2D, with the perturbed flow and
magnetic field perpendicular to the mean field. Along
the field there is an ohmic current and its driving electric
field. The reduced MHD equations are
␳
d 2 ⌽
⫽B 0 ⵜ 储 J 储 ⫹ⵜ⬜ A 储 ⫻z•ⵜ⬜ J 储 ,
ⵜ
dt ⬜ B 0
d
␾
A ⫽⫺B 0 ⵜ 储 ⫺ ␩ J 储 ,
dt 储
B0
(5.6)
(5.7)
where ␳ is the plasma mass density, ⵜ⬜2 ⌽B ⫺1
0 is the total
vorticity, J 储 is the plasma current along the equilibrium
field B 0 , A 储 is the magnetic vector potential of the perturbed field along B 0 , and ␩ is the resistivity. The derivative d/dt includes advection by the equilibrium flow
⫺1
ū(x)⫽⫺B ⫺1
0 ⵜ⌽ 0 ⫻b and fluctuating flow ⫺B 0 ⵜ ␾
⫻b, where b is the unit vector along B 0 . Note that the
left-hand side of Eq. (5.6) is identical in form to the
neutral-fluid case [Eq. (3.2)]; the right-hand side is the
curl of the Lorentz force. The first and second terms of
Ohm’s law are the inductive and electrostatic electric
fields of the Lorentz gauge.
For the Kelvin Helmholtz instability magnetic-field
fluctuations are negligible, allowing terms with A 储 to be
dropped (Chiueh et al., 1986). The parallel current gradient of the vorticity equation couples to resistive dissipation, introducing the stabilizing magnetic-shear effect
discussed above. Dropping the nonlinearity and combining these equations by eliminating J 储 , we obtain the vorticity equation
冉
⳵
⫹ik y ū 共 x 兲
⳵t
⫽
B 20
␳ ␩ L s2
冊冉
冊
␾ ky
⳵2
d 2 ū ␾ k y
2
⫺ik y 2
2 ⫺k y
⳵x
B0
dx B 0
k 2y x 2
␾ ky
B0
,
(5.8)
where Eq. (5.4) and the slab reduction [Eq. (5.5)] have
been used. The last term of Eq. (5.8) is the sink associated with ohmic dissipation. Unlike dissipation in neutral fluids [Eq. (3.2)], which is negligible at high Reynolds number, this sink becomes weak when dissipation
(plasma resistivity) is large, because a weaker parallel
current is induced when resistivity is large. The dissipation is also large when x is large, forming a quadratic
eigenmode potential well. At x⫽0, the location of the
rational surface, the electric field has no component
along B 0 , and there is no parallel current or ohmic dissipation. The effect of dissipation on the instability depends on how the instability eigenmode varies with x. If
the fluctuation must extend to large x in order to draw
free energy from the flow shear, the growth rate will be
reduced. This happens either because the fluctuation
structure is broad or because the mean vorticity peaks at
x⫽0 (fluctuations grow in the region of maximum vorticity). However, even a narrow structure centered on
x⫽0 is stabilized by magnetic shear if ␩ is sufficiently
small or k y is large.
138
P. W. Terry: Suppression of turbulence by sheared flow
Numerical evaluation of Eq. (5.8) for a model flow
profile ū(x)⫽V 0 tanh(x/LE) indicates that the mode is
stabilized for all wave numbers of the system if the dissipative term is larger than the drive term (last term of
the left-hand side) at x⬵L E for k y evaluated at the
minimum poloidal wave number of the system (Chiueh
et al., 1986; Sugama and Wakatani, 1991). Thus stability
requires
冉
B 20 k min
d 2 ū/dx 2
1
⬵ 2⬍
ū
LE
␳ V 0 ␩ L s2
冊
1/2
⫽ 共 Lu兲 1/2共 Al兲 1/2a ⫺1 L s⫺1 ,
(5.9)
where Lu⫽ ␮ 0 a v A / ␩ is the Lundquist number, Al
⫽ v A /V 0 is the Alfvén number, and k min is taken to be
the inverse minor radius a ⫺1 . This condition is cited by
Burrell (1997), but V 0 k min is replaced by ⌬ ␻ D , the turbulence decorrelation rate. Equation (5.9) omits a multiplier of order unity and thus is not a precise threshold
criterion, but it does identify the vicinity of the instability threshold. Because the stabilizing dissipative term of
Eq. (5.8) is quadratic in x, it dominates the driving term
at large x. A necessary condition for instability is that
the stabilizing term become weak at some value of x,
i.e., that S M ⫽B 20 k y x 2 / ␳ ␩ L s2 ⫺d 2 ū/dx 2 change sign in
the domain of x. This is equivalent to the instability criterion of Eq. (3.4), which for barotropic instability requires that S R ⫽ ␤ ⫺ ⳵ 2 ū/ ⳵ y 2 change sign in the flow domain (the meanings of x and y are exchanged in the
standard notations of plasma and atmospheric literature). Comparison of these criteria indicates that the
planetary rotation gradient ␤ and magnetic shear
B 20 k y x 2 / ␳ ␩ L s2 play comparable roles in stabilizing flow
shear instability. For typical tokamak parameters, Eq.
(5.9) is satisfied and the Kelvin-Helmholtz modes are
stable.
2. Instabilities in a nonuniform magnetic field
In Sec. II. C it was pointed out that flow shear can
stabilize certain plasma collective modes. The criterion
of Eq. (2.33) is too simple to provide a predictive stabilization threshold, in large part because it fails to account for inhomogeneities such as magnetic shear. In the
presence of magnetic shear, each unstable mode has a
different eigenmode structure, resulting in a different interaction with the flow shear inhomogeneity. Hence, in
contrast to turbulent decorrelation [Eq. (2.17)], there is
no single criterion for the effect of flow shear on linear
stability. In this subsection the effect of flow shear on
several instabilities is described.
The drift wave is an electrostatic fluctuation involving
perturbations of the flow and electron density that is
closely related to the atmospheric Rossby wave. Advection of the mean density by the perturbed E⫻B flow
induces wave propagation at a rate known as the diamagnetic flow speed. In a neutrally stable drift wave, the
rapid motion of electrons along the magnetic-field line
enforces a Boltzmann condition, n e ⫽n 0 e ␾ /T e , making
the perturbed density proportional to the potential. DisRev. Mod. Phys., Vol. 72, No. 1, January 2000
sipative electron effects [e.g., Landau damping (Krall
and Trivelpiece, 1973), other resonances, and collisions]
shift the phase of the density perturbations relative to
the potential and destabilize the wave. Dissipation of ion
motion along the field lines (e.g., Landau damping and
viscosity) damps wave motion and opposes the instability. A basic model for unstable drift waves in the presence of flow shear and magnetic shear in a slab geometry is given by Carreras et al. (1992):
d
⳵␾
⳵ 2␾
⫹D 0 2
共 1⫺ ␳ s2 ⵜ⬜2 兲 ␾ ⫹U D
dt
⳵y
⳵y
冋 冉 冊 册
⫺L n D 0 ⵜ⬜
C s2 2
⳵␾
⫻z •ⵜ⬜ ␾ ⫺
ⵜ ␾ ⫽0. (5.10)
⳵y
vi 储
In Eq. (5.10), d/dt⫽ ⳵ / ⳵ t⫺B ⫺1
0 ⵜ⌽⫻z•ⵜ is the advective
derivative of the equilibrium E⫻B flow, ␳ s
⫽(T e m i ) 1/2/eB 0 is the ion radius of gyration in the magnetic field (evaluated at the electron temperature), C s
⫽(T e /m i ) 1/2 is the ion sound speed, U D ⫽C s ␳ s L n⫺1 is
the diamagnetic flow speed, L n⫺1 ⫽n ⫺1
0 dn 0 /dx is the inverse scale length of the equilibrium density, D 0 is the
strength of the electron dissipation coupling, producing
both the destabilizing term D 0 ⳵ 2 ␾ / ⳵ y 2 and the nonlinearity (Terry and Horton, 1982), v i is the collisional dissipation rate of parallel ion momentum, and ␾ has been
normalized by T e / 兩 e 兩 (T e is the electron temperature
times the Boltzmann constant). The flow profile is assumed to be stable; hence the curvature term is neglected. The first and last terms are almost identical to
the first and last terms of Eq. (5.8). Here there is the
addition of d ␾ /dt, from dn e /dt (with n e ⫽n o e ␾ /T e ),
and the magnetic-shear damping term arises from parallel ion motion instead of parallel current. (Parallel momentum is modeled by the equation n o ␷ i u 储 ⫽⫺C s2 ⵜ 储 n i ,
describing the balance of collisional drag with parallel
ion pressure. This equation couples to ion momentum
through parallel compression of the flow.)
Under linearization and a Fourier transform in y and
t, Eq. (5.10) becomes an eigenmode equation in x, the
direction of inhomogeneity of both the mean flow and
the parallel damping term. In the absence of the mean
flow, the least stable eigenmode peaks about the rational
surface with a width that is determined from the balance
of the time derivative of vorticity and the parallel damping
term,
yielding
⌬⫽( ␻␳ s2 ␷ i L s2 /C s2 k 2y ) 1/4
1/2
1/2
⬵ ␳ s (L s /L n ) ( ␷ i /k y U D ) . With linear flow shear,
⫺B ⫺1
0 ⵜ⌽⫻z⫽U y (x)⫽U y (0)⫹xU y⬘ (0), the peak of the
eigenmode structure is shifted off the rational surface by
the linear inhomogeneity of the flow shear. The shift S is
proportional to the shear strain rate ␻ s normalized to
the drift-wave propagation frequency Doppler shifted
by the mean flow speed at the rational surface, i.e., S
The
shift
⫽k y U y⬘ (0)⌬/ 关 ␻ ⫺k y U y (0) 兴 ⬵ ␻ s /k y U D .
forces the eigenmode into a region of stronger parallel
damping, causing a reduction of growth rate. In the expression for the growth rate, flow shear enters as an additive reduction to the linear instability drive. Complete
P. W. Terry: Suppression of turbulence by sheared flow
stabilization occurs when the shear term is larger than
the difference of the drive term and the parallel damping term:
␻ s ⬎2k y U D
冋
␳ s2
␳ s k 2y D 0
⫺
W k y U D 冑 2W 2
册
1/2
,
(5.11)
where W is the mode width ⌬ evaluated at ␻ ⫽k y U(0)
⫹k y U D . The factor ␳ s /W⬃(L n /L s ) 1/2 and the term in
brackets are both considerably less than unity, indicating
that the mode is stabilized at a flow shear strength far
below the linear stability estimate of Eq. (2.33), which
yields ␻ s ⬎k y U D . However, Eq. (5.11) is comparable to
the nonlinear suppression criterion, Eq. (2.17), because
saturated turbulence adjusts to make the turbulence
decorrelation rate essentially equal to the growth rate,
particularly for long wavelengths (k y ␳ s ⬍1) where there
is little dispersion.
Ion-temperature-gradient modes are a drift-wave instability believed to be an important source of turbulence in tokamaks. Free energy for the instability is provided by the ion temperature gradient. At the instability
threshold, where the temperature gradient scale length
just exceeds a critical value, the gradient free energy is
accessed through a resonance of the mode with the
guiding-center motion of individual ions under the drifts
produced by the gradient and curvature of the magnetic
field. Correct modeling of the threshold requires kinetic
theory. Well above the instability threshold, the mode
frequency becomes larger than the frequency of drift
motion, and fluid equations can be used to model the
instability and the turbulence it drives (Lee and Diamond, 1987). The spatial structure of the fluctuations
has different forms in a slab and a torus. Moreover, in
toroidal geometry, there are two branches, one whose
structure is like that of a slab, and one whose structure is
uniquely toroidal, with a tendency for the fluctuations to
be much larger at ␪ ⫽0 than at ␪ ⫽ ␲ . Because of these
and other complexities, there are numerous models and
growth-rate calculations for ion-temperature-gradient
modes (for a review, see Horton, 1999).
In the ion-temperature-gradient mode, flow shear is
destabilizing at weak levels and becomes stabilizing
when it is stronger. This feature is related to the effective eigenmode potential and is most easily illustrated
using a fluid model (Wang, Diamond, and Rosenbluth,
1992). In addition to vorticity and parallel flow equations, the fluid model has an ion pressure evolution
equation,
冋 冉 冊 册
d
1⫹ ␩ i 2 ⳵ ␾
ⵜ⬜
⫹ⵜ 储 u 储 ⫽0,
共 1⫺ⵜ⬜2 兲 ␾ ⫹U D 1⫹
dt
T
⳵y
(5.12)
d
u ⫽⫺ⵜ 储 ␾ ⫺ⵜ 储 p,
dt 储
(5.13)
1⫹ ␩ i ⳵ ␾ ⌫
d
p⫹U D
⫹ ⵜ u ⫽0,
dt
T
⳵y T 储 储
(5.14)
冉 冊
where ␾, u 储 , and p are normalized by T e /e, C s , and
n o T e , spatial scales are normalized by ␳ s , T⫽T e /T i ,
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
139
␩ i ⫽d ln Ti /d ln no , and ⌫ is the ratio of specific heats.
Under linearization, the derivative d/dt includes advection by the equilibrium flow ū(x)⫽⫺B ⫺1
0 ⵜ⌽ 0 ⫻z. The
drift terms (those with U D dependence) come from E
⫻B advection of the equilibrium pressure in the ion momentum and pressure evolution. When ␩ i ⬎1 the instability is driven primarily by the temperature gradient
contribution to the pressure gradient.
In Eq. (5.12) the parallel compression term ⵜ 储 u 储
couples the dynamics of parallel ion motion to the vorticity. Unlike the prior example, in which the parallel
motion was purely dissipative, here parallel motion is
driven by the pressure, which through the advection of
the mean pressure gradient drives the instability. Consequently larger values of ⵜ 2储 away from the rational surface produce stronger drive. However, this tendency is
limited by the parallel electric field in Eq. (5.13), which
acts as a dissipative sink in this model. At sufficiently
large distances from the rational surface this dissipation
overcomes the drive, and the fluctuation structure becomes evanescent. This feature is evident in the behavior of the growth rate as a function of the radial eigenmode number in the ū⫽0 case. The eigenmode number
is a measure of the rms displacement from the rational
surface. With increasing eigenmode number, the growth
rate first increases as the coupling to the pressure drive
becomes stronger, and then decreases as the dissipation
associated with the parallel electric field overcomes the
gradient drive (Terry et al., 1988).
When flow shear with a linear profile is added to the
equations, it shifts the fluctuation structure further from
the rational surface. Thus weak shear can enhance the
coupling to the pressure drive and thereby increase the
growth rate, while stronger shear favors compressional
damping and lowers the growth rate (Hamaguchi and
Horton, 1992; Wang, Diamond, and Rosenbluth, 1992).
Strong shear also introduces a wave absorption resonance that further damps fluctuations (Wang, Diamond,
and Rosenbluth, 1992). Kinetic treatments of the iontemperature-gradient instability in slab geometry show
the same behavior, with a growth rate that increases
with weak shear and decreases with stronger shear
(Staebler and Dominguez, 1991; Wang, Diamond, and
Rosenbluth, 1992). In toroidal geometry, flow shear
strongly affects eigenmodes because of their large radial
extent (Connor, Taylor, and Wilson, 1993). In simulation of ion-temperature-gradient turbulence in toroidal
geometry the shear flow is typically generated selfconsistently by the Reynolds stress. When the shear
strain rate is larger than the linear growth rate, there is a
significant reduction of turbulence and a tenfold reduction of transport (Waltz, Kerbel, and Milovich, 1994).
The effect of shear flow on the ion-temperature-gradient
mode has also been investigated experimentally in a linear plasma device (Song and Sen, 1994). This mode,
which appears as a quasicoherent feature in the fluctuation spectrum, was observed to increase in amplitude
with weak shear and decrease with stronger shear. This
behavior was attributed to the fact that the same external control, i.e., the voltage on an rf heating grid, affects
140
P. W. Terry: Suppression of turbulence by sheared flow
both the ion-temperature-gradient drive and the flow
shear. It was postulated that the rf voltage favors the
ion-temperature-gradient drive at lower values and the
generation of sheared flow at higher values. Complete
stabilization of the mode required considerably larger
flow shear than the theoretically predicted values.
The resistive interchange instability is a pressuredriven interchange mode that is closely related to the
Rayleigh-Taylor-instability. It is unstable when the pressure gradient is parallel to the curvature of the equilibrium magnetic field. The field-line curvature therefore
acts as an effective gravity. The instability is described
by the reduced MHD equations, Eqs. (5.6) and (5.7),
and is damped by the dissipation of parallel current excited by the flow, just as with the Kelvin-Helmholtz instability. Pressure couples to the vorticity evolution
through the pressure interchange torque (Sugama and
Wakatani, 1991), ␬ ⫻ⵜp•z, where ␬ is the curvature of
magnetic-field lines. With this term the vorticity equation is
␳
d 2 ⌽
⳵p
ⵜ
.
⫽B 0 ⵜ 储 J 储 ⫺ ␬
dt ⬜ B 0
⳵y
(5.15)
As with the Kelvin-Helmholtz instability, the interchange mode is dominantly electrostatic, allowing the
Lorentz term involving the inductive field A 储 to be
dropped. The parallel current is governed by the righthand side of Ohm’s law, Eq. (5.7). A simple pressure
evolution equation, dp/dt⫽⫺(dp 0 /dr) ⳵ / ⳵ y( ␾ /B 0 ), describing advection of perturbed pressure by the equilibrium flow and advection of the equilibrium pressure by
the perturbed flow, closes the system. The linear eigenmode equation is like the Kelvin-Helmholtz eigenmode
equation, Eq. (5.8), with the addition of the curvature
drive:
关 ⫺i ␻ ⫹ik y ū 共 x 兲兴
冉
冊
␾ ky
⳵2
d 2 ū ␾ k y
2
⫺k
⫺ik
y
y
⳵x2
B0
dx 2 B 0
␾ k y dp 0
␬ k 2y
2 2
,
⫽
k
x
⫺
2 y
B0
␳ 关 ⫺i ␻ ⫺ik y ū 共 x 兲兴 B 0 dr
␳␩Ls
B 20
␾ ky
(5.16)
where the temporal Fourier transform has been introduced. The advective pressure response in the denominator of the curvature drive complicates the eigenmode
structure; hence a simple characterization of the effect
of flow shear is not possible. Clearly large flow shear
weakens the curvature drive, but large shear flow also
triggers the Kelvin-Helmnoltz instability. There is a regime where flow shear stabilizes the resistive interchange instability, but the system is Kelvin-Helmnoltz
stable (Sugama and Wakatani, 1991; Tajima et al., 1991;
Carreras et al., 1993). For smaller resistivity, KelvinHelmholtz stability is possible with stronger flow shear,
and the resistive interchange mode is stable at larger
values of the curvature. The real part of the stabilized
interchange eigenmode remains centered at the rational
surface, but the eigenfunction develops a significant
imaginary part (Sugama and Wakatani, 1991). The
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
eigenfunction can be modeled as a shifted Gaussian with
an imaginary shift proportional to the flow shear magnitude (Carreras et al., 1995).
The above examples demonstrate that strong (KelvinHelmholtz-stable) flow shear frequently stabilizes collective plasma instabilities. Unlike the nonlinear decorrelation process of Sec. II, the details and mechanism vary
from case to case. Hence there is no universal criterion
for stabilization, and in some cases flow shear, particularly weak flow shear, can be destabilizing. Moreover,
the ideal of a collective linear mode in the presence of a
prescribed flow shear, though convenient for analysis
and interpretation, is not generally realized in plasmas.
A variety of numerical studies in which the fluctuations
are allowed to act on the shear flow through the mechanisms of Sec. IV show that there is a strong feedback,
and the effect of flow shear on fluctuations must be
treated self-consistently (Carreras, Lynch, and Garcia,
1991; Ware et al., 1992; Guzdar et al., 1993; Waltz, Kerbel, and Milovich, 1994). The feedback mechanisms of
these simulations are nonlinear, also indicating that the
effect of flow shear on fluctuations likely involves some
combination of the physics of linear stabilization and
nonlinear decorrelation.
3. Shear straining in a torus
Observations of fluctuation suppression in tokamaks
are consistent with the shear suppression criterion, Eq.
(2.17), provided it is modified for the tokamak geometry. In this section, the E⫻B shearing rate appropriate
for tokamak fluctuations is derived. Consider a sheared
poloidal flow driven by the E⫻B drift of a radial electric
field in a torus. For the Fourier transform of toroidal
and poloidal angles introduced just prior to Eq. (5.2),
the rate of poloidal advection is
␻ E⫻B ⫽⫺
E r im
.
B␸ r
(5.17)
A wide class of fluctuations are sufficiently localized in
the vicinity of the rational surface that the expression
m⫽qn, which is precisely true only on a rational surface, remains approximately true over the radial extent
of the fluctuation. Using this and the definition of q in
Eq. (5.3), we obtain the advection rate ␻ E⫻B
⫽⫺in(E r /RB ␪ ). The shear straining rate is the difference of advection rates at two radial locations separated
by a radial correlation length ⌬r,
␻ s⫽
冉 冊
⌬r ⳵ E r
,
⌬ ␸ˆ ⳵ r RB ␪
(5.18)
where ⌬ ␸ˆ ⫽⌬ ␸ /2␲ ⫽(in) ⫺1 is the toroidal correlation
length (wavelength) normalized by 2␲. The difference
has been expanded in a Taylor series, retaining only the
first term. From Eq. (5.18) the shearing rate depends on
variations of both the radial electric field and the poloidal magnetic field.
In a torus, the flux surfaces need not have circular
cross sections, and a radial coordinate that uniquely
141
P. W. Terry: Suppression of turbulence by sheared flow
identifies individual flux surfaces is advantageous. Such
a coordinate can be defined from the poloidal flux
through a toroidal annulus of width dr and radius 2 ␲ R
in the toroidal symmetry plane,
d ␺ ⫽2 ␲ RB ␪ dr,
(5.19)
where ␺ is the flux coordinate. The toroidal symmetry
plane is the plane of ␪ ⫽0 and ␪ ⫽ ␲ , i.e., the plane in
which a bagel is sliced. A flux coordinate normalized by
2␲ is often used: ␺ˆ ⫽ ␺ /2␲ . In terms of the normalized
flux coordinate, ⌬r ⳵ / ⳵ r→⌬ ␺ˆ ⳵ / ⳵ ␺ˆ , and the shearing rate
is
␻ s⫽
⌬ ␺ˆ ⳵
冉 冊
Er
⌬ ␸ˆ ⳵ ␺ˆ RB ␪
.
(5.20)
This is the expression given by Burrell (1997) for the
shearing rate in a tokamak. It is general, despite our use
in the present context of the circular form of q [Eq.
(5.3)]. An equivalent expression is given by Hahm and
Burrell (1995) by transforming E r ⫽⫺ ⳵ ⌽ 0 / ⳵ r to flux coordinates.
Under
this
transformation,
Er
⫽( ⳵ ⌽ 0 / ⳵ ␺ˆ )d ␺ˆ /dr⫽( ⳵ ⌽ 0 / ⳵ ␺ˆ )RB ␪ , and
␻ s⫽
⌬ ␺ˆ ⳵ 2
⌬ ␸ˆ ⳵ ␺ˆ 2
共 ⌽0兲.
(5.21)
Since the mean electrostatic potential ⌽ 0 is constant on
a flux surface, the shear parameter of Eq. (5.20),
⳵ / ⳵ ␺ˆ (E r /RB ␪ ), is also constant on a flux surface. Isolating which parts of an expression vary over a flux surface
and which parts do not is important in addressing the
origin of asymmetries with respect to ␪. In the case of
Eqs. (5.20) and (5.21), asymmetries in poloidal angle
originate with the ratio of correlations ⌬ ␺ˆ /⌬ ␸ˆ .
This ratio can be expressed in terms of the magnetic
field and the major radius R (Burrell, 1997) as follows:
From the flux coordinate definition, Eq. (5.19), ⌬ ␺ˆ is
related to a radial correlation by ⌬r⫽(B ␪ R) ⫺1 ⌬ ␺ˆ . The
toroidal correlation ⌬ ␸ˆ is related to the correlation
length L⬜ perpendicular to the magnetic field in a flux
surface. The latter is determined from the perpendicular
gradient in a flux surface, L⬜⫺2 ⫽ 兩 ␺ˆ ⫺1 ⵜ ␺ˆ ⫻ⵜ⬜ 兩
⫽ 兩 ␺ˆ ⫺1 ⵜ ␺ˆ ⫻(ⵜ⫺B ⫺1 B•ⵜ) 兩 . In terms of the Fourier
transform of toroidal and poloidal angles,
L⬜⫺1 ⫽
冏 冉
冊
冉
冊冏
in
B␸
im
B␪
1⫺
␸ˆ ⫹
1⫺
␪ˆ ,
R
B
r
B
(5.22)
where ␸ˆ and ␪ˆ are unit vectors in the toroidal and poloidal angles, respectively. In a tokamak the toroidal
field component is an order of magnitude larger than the
poloidal component. Therefore the second term of Eq.
(5.22) dominates and L⬜⫺1 ⬵m/r⫽nq/r⫽B ␸ /⌬ ␸ˆ RB ␪
⬵B/⌬ ␸ˆ RB ␪ . Observations (Fonck et al., 1992) and
simulations (Parker, Lee, and Santoro, 1993; Waltz, Kerbel, and Milovich, 1994) indicate that to a good approximation ⌬r⬵L⬜ , from which
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
⌬ ␺ˆ R 2 B 2␪
⬵
.
⌬ ␸ˆ
B
(5.23)
Substituting Eq. (5.23) into the shear strain rate, Eq.
(5.20) yields
␻ s⫽
R 2 B ␪2 ⳵
B
冉 冊
Er
⳵ ␺ˆ RB ␪
.
(5.24)
In this expression, the factor R 2 B ␪2 /B is not a constant
along the flux surface. The magnitude of B varies as R ⫺1
and R⫽R 0 ⫹r cos ␪ varies significantly from ␪ ⫽0 to ␪
⫽␲.
A variation of the shear straining rate from the outside midplane ( ␪ ⫽0) to the inside midplane ( ␪ ⫽ ␲ ) is
consistent with observations of fluctuations in the Doublet III-D (DIII-D) tokamak in the H mode (Burrell
et al., 1992; Doyle et al., 1992) and for a core transport
barrier (Burrell, 1997). In the latter the shear straining
rate is calculated to increase by a factor of 7 from the
inside midplane to the outside midplane. Scattered light
from a far-infrared laser diagnostic indicates that density
fluctuations are larger at the inside midplane than at the
outside midplane. This observation is striking because
the outside midplane is a region of unfavorable
magnetic-field curvature and therefore enhanced susceptibility to interchange-driven turbulence. In the absence
of flow shear, the outside midplane generally has stronger fluctuation activity. This poloidal asymmetry is also
reflected in the spatial and temporal behavior of the
transport barrier induced by biased probes. The rise in
the total number of electrons and electron kinetic energy associated with flow-shear-induced suppression
commences at the outside midplane (Jachmich et al.,
1998). The factor E r /RB ␪ in the derivative is also significant in understanding other observations. In a type of
enhanced confinement mode called the VH mode, the
H-mode transport barrier broadens inward to include a
larger plasma volume, as indicated by the region of fluctuation reduction and steepened pressure gradient. In
some cases the confinement improvement covers all of
the plasma volume. Shear in the E⫻B velocity E r /B is
significant only in the outer part of the region of improved confinement, whereas shear in E r /RB ␪ extends
over the entire region of improved confinement (Lao
et al., 1996).
B. Effect of flow shear on transport fluxes
The fluctuation-induced transport fluxes of density
and heat, Eqs. (2.8) and (2.9), depend on the amplitudes
˜ Àk , p̃ k , and ñ k , but they also
of turbulent fluctuations ␾
depend on the complex phase angle between each of the
two fluctuations that compose the correlation. In terms
of a phase representation of the complex Fourier fluctuation amplitudes, the heat flux can be written
Re Q e ⫽⫺B ⫺1
0
兺k k y兩 ␾˜ ⫺k兩兩 p̃ k兩 sin ␣ k⌰ k ,
(5.25)
142
P. W. Terry: Suppression of turbulence by sheared flow
where the cross phase ␣ k is defined as the difference of
˜ ⫺k and p̃ k in the complex plane,
the phase angles of ␾
exp共 i ␣ k兲 ⬅
˜ ⫺kp̃ k典
具␾
˜ ⫺k兩兩 p̃ k兩
兩␾
,
(5.26)
and ⌰ k is the coherency (Bendat and Piersol, 1986). The
˜ ⫺kp̃ k兩 兵 兩 ␾
˜ k兩 2 兩 p̃ k兩 2 其 ⫺1/2, is a
coherency, defined by ⌰ k⫽ 兩 ␾
measure of noise. If the fluctuations are perfectly correlated but out of phase, ⌰ k⫽1 and ␣ k⫽ ␲ . We have discussed in Sec. II. B the effect of flow shear on turbulence
amplitudes. Here its effect on the cross phase ␣ k is discussed.
A simple illustration of the effect of flow shear on the
cross phase can be drawn from the pressure evolution
equation of the resistive interchange instability, given
just prior to Eq. (5.16). If there is no flow, linearization
yields
p k⫽
␾ k k y dp 0
.
B 0 ␻ dr
(5.27)
The unstable root of the dispersion relation has a purely
imaginary frequency (growth rate),
冉
␬ 2 k 2y L s2 ␩ 共 dp 0 /dr 兲 2
␻ ⫽i ␥ k ⫽i
B 20 ␳
冊
1/3
.
(5.28)
Therefore, for the linear instability, (k y / ␻ )⫽⫺ik y / ␥ k ,
˜ k兩 2 关 k 4y (dp 0 /
and ␣ k⫽⫺ ␲ /2. The flux Re Qe⫽⌺k兩␾
⫺4 ⫺2 ⫺2 ⫺1 1/3
dr) ␳ B 0 ␬ L S ␩ 兴 is positive, and heat is transported outward (the coherency ⌰ k has been ignored). If
there is an equilibrium flow and advection dominates
the pressure response (k y U y Ⰷ ␻ ), the pressure is given
by
p k⫽⫺
␾k
ky
dp 0
.
B 0 k y U y 共 x 兲 dr
(5.29)
If the frequency of advection k y U y is real, the crossphase angle is ␣ k⫽ ␲ , and there is zero transport. This
illustration only suggests how flow affects the cross
phase because it relies on gross simplifications. The flow
U y (x) extends from the rational surface, and the corre˜ ⫺kp̃ k典 therefore samples the eigenmode. Belation 具 ␾
cause the eigenmode becomes complex in the presence
of flow shear, the phase angle is neither ␲ nor ␲/2.
Moreover, at finite amplitude, the nonlinear pressure response must also be considered, and the eigenmode itself is nonlinear. These effects decrease the phase factor
sin ␣ k from its maximum value of unity in the case of the
linearly unstable mode, but quantifying the decrease is
difficult and requires approximation.
Analytic studies of the effects of flow shear on the
cross phase have been largely limited to resistive interchange turbulence. The first discussion (Carreras et al.,
1995) included the turbulent diffusivity D in the advection of pressure, representing the pressure equation
as 关 ik y U y (x)⫺D ⳵ 2 / ⳵ x 2 兴 p k⫽⫺ik y B ⫺1
0 ␾ kdp 0 /dr. This
equation was inverted heuristically,
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
Im
ik y
关 ik y U y 共 x 兲 ⫺D ⳵ 2 / ⳵ x 2 兴
→
⫺D/2W 2
⫽sin ␣ k ,
共 k 2y U 2y ⫹D 2 /4W 4 兲 1/2
(5.30)
accounting for the eigenmode width W in the turbulent
diffusion, but not accounting for the shifted, complex
eigenmode structure in the form of the flow shear term.
A subsequent study (Ware et al., 1996) dealt with the
nonlinear eigenmode problem in a more consistent fashion. This study also considered a renormalized model
(Carreras, Garcia, and Diamond, 1987), i.e., one in
which the advection of vorticity and pressure was replaced by an amplitude-dependent viscous diffusion operator and a turbulent pressure diffusivity, respectively.
The nonlinear eigenmode problem was solved approximately, using an eigenfunction ansatz with an undetermined mode width and imaginary shift. Moments were
taken in order to determine the mode width, the shift,
the magnitudes of the turbulent viscosity and diffusivity,
and the appropriate form of the operator inversion in
Eq. (5.30). The cross phase in the weak shear limit was
found to be
冉
sin ␣ k⫽⫺ 1⫺
5k 2y U ⬘y 共 0 兲 2 ⌬ k6
4D 2
冊
,
(5.31)
where D is the turbulent pressure diffusivity, U y⬘ (0) is
the flow shear rate at the rational surface, ⌬ k
⫽W( ␮ /W 2 ␥ k ) 1/6 is the nonlinear mode width, ␥ k is the
linear growth rate [Eq. (5.28)], ␮ is the turbulent viscos⫺2 1/6
is the linear
ity, and W⫽ 关 ␳ ␩ 2 L s4 ␬ (dp 0 /dr)B ⫺4
0 ky 兴
mode width. Both Eqs. (5.30) and (5.31) indicate that in
the limit of no flow, sin ␣ k⫽⫺1, as in the case of the
linear instability with no flow, while nonzero flow decreases the value of sin ␣ k , independent of any amplitude effects. Both results yield a reduction of cross
phase, independent of the sign of the flow shear. Flow
curvature has also been examined in the weak limit
(Kelvin-Helmholtz stable) and was found to reduce the
cross phase (Ware et al., 1996).
An extension of the resistive interchange model shows
that in circumstances in which the conduction of heat
along the magnetic field is important in the turbulent
dynamics, shear flow can lead to a marked decrease in
the particle flux with almost no reduction in the heat flux
(Ware et al., 1998). The effect is sensitively dependent
on the relationship between flow shear and the cross
phase through the inversion of the relevant advective
operators, indicated heuristically by Eq. (5.30). To address both heat and particle transport the pressure equation is split into two equations, one for density and one
for temperature. Temperature fluctuations are subject to
heat conduction along the magnetic field, carried by rapidly streaming electrons. The parallel thermal conductivity goes as the square of the electron thermal speed divided by the electron collision frequency. In the density
equation, parallel dissipation involves the propagation
of sound waves, with an effective diffusivity that scales
as the square of the sound speed divided by the ion
P. W. Terry: Suppression of turbulence by sheared flow
collision rate. This diffusivity is smaller than the parallel
thermal conductivity by the square root of the electronto-ion mass ratio and is easily smaller than the linear
and nonlinear time scales in the density evolution. The
same is not true for the temperature evolution. In the
edge of fusion plasmas it is possible to have a regime in
which the temperature evolution, 关 U y (x) ⳵ / ⳵ y⫺u•ⵜ
⫺ ␹ 储 ⵜ 2储 兴 T⫽⫺B ⫺1
0 ⳵ ␾ / ⳵ ydT 0 /dr, is dominated by the
balance of the thermal conduction with the advection of
the mean temperature, even in the presence of a mean
shear flow. Under such a circumstance there is no effect
of flow shear on the cross phase. In the density equation,
关 U y (x) ⳵ / ⳵ y⫺u•ⵜ 兴 n⫽⫺B ⫺1
0 ⳵ ␾ / ⳵ ydn 0 /dr, flow shear
diminishes cross phase as discussed in the previous paragraph. This effect provides an explanation for the observed behavior of heat and particle transport in edge
shear flows in the Continuous Current Tokamak (Tynan
et al., 1996) and the Madison Symmetric Torus
Reversed-Field Pinch (Craig et al., 1997). In both cases,
shear flows induced by biased probes produced a
marked decrease in particle transport with virtually no
change in the heat flux. The heat and particle fluxes calculated from the model for the parameters of these devices show the same features (Ware et al., 1998). In contrast, the edge plasmas of the TEXTOR tokamak have a
higher mean density, leading to a parallel thermal conduction rate that is smaller than the turbulence decorrelation rate and shear strain rate. There, a reduction of
both the temperature fluctuations and the heat flux is
observed (Boedo et al., 1998).
Measuring the cross phase is difficult because two
fluctuations must be measured simultaneously in the
same spatial location. Such measurements are most frequently carried out with Langmuir probes, from which
the fluctuations in plasma density and potential can be
inferred. Langmuir probe measurements do not directly
˜ k . They remeasure the fluctuating plasma potential ␾
quire assumptions about temperature fluctuations,
which are often not measured, in order to infer the potential fluctuations. This limitation should be borne in
mind in assessing Langmuir probe data. A reciprocating
Langmuir probe has been employed to measure the
cross phase and fluctuation amplitudes of the particle
flux in DIII-D (Moyer et al., 1995). To accommodate a
probe, colder pre-H-mode and H-mode plasmas were
produced by limiting auxiliary heating to a short neutral
beam pulse that triggered the H phase. Several aspects
of these measurements are of interest. The particle flux
is observed to decrease much more dramatically than
the fluctuations, which remain at finite amplitude. The
flux is reduced by 1–2 orders of magnitude over a range
of radial positions in which the flow shear varies from
strongly positive to strongly negative. There is no peaking of the particle flux even where the flow shear goes
through zero. In qualitative agreement with shear sup˜ ⫺k and ñ k , normalized to
pression, the amplitudes ␾
their pre-H-mode values, are larger in the region of zero
flow shear than outside it, where shear is large. The fact
that the flux remains small suggests that the cross phase
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
143
is near an integer multiple of ␲. Indeed, a direct measurement of the phase angle at this location reveals that
␣ k changes from ␲/2 to ⫺␲ in going from the pre-Hmode to the H mode. This establishes that the cross
phase can be modified independently of the amplitudes.
It also suggests that the cross phase is sensitive to features of the flow other than its shear, e.g., the flow curvature. The cross phase has also been measured in an H
mode induced by external biasing, again using Langmuir
probes (Boedo et al., 1998). In the region where the flow
shear changes sign, the cross phase causes the particle
flux to shift from outward to inward.
VI. FLOW SHEAR AND ENHANCED PLASMA
CONFINEMENT
The importance of confinement in fusion devices has
encouraged the discovery of numerous modes of operation with improved confinement (Carreras, 1997).
Strong flow shear is present in those enhancedconfinement modes that have a universal character, i.e.,
that are reproducible under diverse conditions and occur
in a large variety of distinct magnetic confinement configurations. It is also conjectured to be present in other
enhanced-confinement modes (Carreras, 1997). Although observational details remain to be understood,
particularly those that relate to how transport barriers
are generated, flow-shear-induced suppression of turbulence and transport is the leading explanation for the
improvement of confinement. This follows from the observed temporal and spatial coincidence of transport reduction with the region of strong flow shear; from numerous features of the reduction that are in agreement
with theory, as discussed in previous sections; and, importantly, from the requisite universality of the flow
shear suppression mechanism. Other explanations have
lacked this universality. This section reviews the salient
features of enhanced confinement in fusion devices.
A. Tokamak
The most extensive body of knowledge on enhancedconfinement operation comes from the tokamak. In this
device, transport barriers are either formed spontaneously or induced by external means. Moreover, they occur in both the core and the edge regions of the plasma.
1. Spontaneous edge transport barrier
The spontaneous formation of a transport barrier in
the plasma edge is generally regarded as a transition to
the H mode. The H mode is remarkable for its universality. Every auxiliary heated divertor tokamak since
1982 has reproducibly created H modes (auxiliary heating and divertors are discussed in subsequent paragraphs). References to the extensive number of diverted
tokamaks reporting H mode can be found in the reviews
of Stambaugh et al. (1990), Groebner (1993), and Burrell (1997). The H mode has also been achieved in other
tokamaks and in nontokamak magnetic confinement devices. Other modes of tokamak operation involving en-
144
P. W. Terry: Suppression of turbulence by sheared flow
hanced confinement in conjunction with edge processes,
such as radiation, have been reported. These are peculiar to individual machines (Lazarus et al., 1985; Ongena
et al., 1993; Messiaen et al., 1997) and thus lack the universality of the H mode.
The general properties of the H mode, outlined in
Sec. II. D, occur in all discharges and devices, providing
an operational definition of the H mode. The underlying
physical process is robust, and while affected by the
complexities and highly variable conditions of the
plasma edge, it transcends these conditions as a universal phenomenon. The universal aspects of the H mode
are
(1) The requirement of heating power to the plasma
above a threshold. Heating power increases the heat flux
through the edge, which has been linked to the transition. It also increases the edge temperature.
(2) The existence of a transition or bifurcation to a
new plasma state as manifested by changes in the profiles of temperature and density and local heat and particle fluxes. The transition occurs when a threshold value
of plasma heating or edge temperature is reached.
(3) The presence of strong E⫻B flow shear.
(4) The other features described in Sec. II. D, such as
confinement time increase and H ␣ decrease.
The H mode occurs over a wide range of plasma conditions. In particular,
(1) The heating power required for the transition can
be provided by any of an assortment of techniques with
widely varying effects on the energy distributions and
temperatures of electrons and ions. The H mode has
been produced with the injection of energetic neutral
beams, a method employed in numerous devices; with
the injection of rf power at the electron cyclotron resonance (Hoshino et al., 1989), the ion cyclotron resonance
(Steinmetz et al., 1987), and the lower hybrid resonance
(Tsuji et al., 1990); and by ohmic heating (Osborne et al.,
1990). Ohmic (resistive) heating is associated with the
production and maintenance of plasma current. (For additional references on heating in the H mode see Burrell, 1997).
(2) While it is desirable to have edge control with
regard to concentrations of impurities and neutrals, radiation, and the recycling of plasma particles through
ionization and charge exchange, a particular form of
control is not required. It can be achieved with either
divertors or limiters, devices that lead to widely differing
magnetic and electric fields in the plasma edge region.
The divertor is a set of external magnetic-field coils that
modifies the topology of the edge magnetic field, producing a singularity in the magnetic shear at the x point
of the separatrix. The divertor channels heat and particle outflows away from the plasma-facing wall, thereby
controlling the concentrations of impurity ion species
and neutrals. In limiter discharges the plasma edge is
manipulated by a material surface protruding inward
from the walls. Limiter discharges tend to have a higher
impurity content at the edge, and/or a higher concentration of neutrals, both of which cool the edge and make
triggering the H mode more difficult. Consequently, diRev. Mod. Phys., Vol. 72, No. 1, January 2000
vertors are used routinely for H-mode plasmas, but the
H mode also has been achieved with limiters (Odajima
et al., 1987; Sengoku et al., 1987; Bush et al., 1990; and
Refs. 90–92 of Burrell, 1997).
(3) The transition power threshold varies from device
to device. It is sensitive to the heating mechanism,
whether a divertor or limiter is used, the location of the
divertor, whether particle drifts are away from or toward
the divertor, and other plasma parameters.
(4) The magnitude of the changes in profiles, confinement time, the time required for the transition to the H
mode, and the other changes described in Sec. II. D all
vary when plasma conditions are changed.
(5) The H mode occurs in the widely different magnetic geometries of nontokamak devices.
The large variability of plasma conditions associated
with the H mode requires a physical mechanism of considerable generality. A variety of proposed models have
been tied to a particular collective instability, geometry,
or field topology. They did not have the requisite generality and failed in experimental comparisons. However,
stable flow shear provides a universal mechanism for
turbulence reduction. This mechanism applies under all
the observed variations of the H mode. The identification of this mechanism does not mean that the H mode
is fully understood. A predictive theory of the H mode
that accounts for generation of shear flow under all of
the variable circumstances of the transition does not at
present exist.
2. Externally induced edge transport barrier
The external biasing technique discussed in Sec.
IV. B. 2 produces an edge transport barrier by externally
driving E⫻B flow shear. These transport barriers share
many common features with the spontaneous H mode,
including a bifurcation of the plasma equilibrium, the
subsequent formation of steep gradients in the region of
flow shear, a decrease in H ␣ emission, and an increase in
plasma energy. Thus the edge transport barrier produced by external biasing is regarded as an induced H
mode. Several aspects of induced H modes are noted
here. Particle confinement is generally observed to increase, indicating a decrease in the particle flux. In some
cases the energy confinement also increases, although
weakly in comparison to the particle confinement (Weynants et al., 1991). In other cases there is virtually no
change in energy confinement (Tynan et al., 1996; Craig
et al., 1997). Differences in the response of heat and particle fluxes to flow shear can be understood if collisional
heat conduction along the magnetic field is important
(see Sec. V. B). Where measured, the fluctuation levels
of density and potential also decrease (Tynan et al.,
1992). The improvement of particle confinement with
biasing can be obtained without the bifurcation (Weynants and Van Oost, 1993). This indicates that favorable
changes in particle confinement are attributable to the
flow shear and not simply changes in equilibrium that
occur with bifurcation. In the reversed-field pinch, a bifurcation with biasing has not been observed (Craig
P. W. Terry: Suppression of turbulence by sheared flow
145
et al., 1997). In the biasing experiments, unlike the spontaneous H mode, the sign of the electric field can be
varied from positive to negative. While the increases in
particle confinement time differ for the two cases, there
is an increase for both signs (Weynants et al., 1991), consistent with shear suppression (Secs. II. B and V. B).
3. Spontaneous internal transport barriers
The existence of several distinct internal transport
barriers has been reported. In many cases internal barriers have led to greater gains in confinement time than
from the H mode alone. In the most notable internal
barrier, ion thermal and particle transport rates fall to
the irreducible minimum arising from collisions in a
nonequilibrium plasma.
The internal barrier that has the most direct connection with the H mode is the VH mode (Jackson et al.,
1991; Greenfield et al., 1993). The designation VH is for
‘‘very high,’’ a reference to the enhancement of confinement time by a factor of 2 beyond that of the H mode, as
observed in DIII-D. The VH mode occurs as an inward
expansion of the region of steepened density and temperature gradients, coinciding with an inward penetration of the region of strong flow shear, decreased ion
thermal conductivity, and decreased density fluctuations.
In the Joint European Torus (JET), the toroidal flow
and its radial shear are observed to increase throughout
the VH mode. The VH mode requires a quiescent H
mode, free of coherent edge fluctuations known as edgelocalized modes (ELM’s) that sometimes appear in the
H mode. More recently, internal transport barriers with
no connection to the H mode have been reported. In the
JT-60U tokamak, a barrier forms spontaneously at an
internal radial location, as manifested by a marked
steepening of the gradients of ion temperature and electron density and the inferred decrease in the local ion
thermal diffusivity (Koide et al., 1994). The barrier formation temporally coincides with a large increase in the
E⫻B flow due to poloidal rotation. This barrier is not
created as an extension of the H mode. On the contrary,
MHD activity driven at the barrier by a steepened pressure gradient can produce a heat pulse that propagates
to the edge and triggers an H-mode transition.
The most impressive transport barrier from the standpoint of its effect on global confinement is a barrier that
links favorable magnetic-shear configurations (Hugon
et al., 1992) with flow shear suppression. This barrier
(Levinton et al., 1995; Strait et al., 1995; Koide et al.,
1997) has come to be referred to simply as an internal
transport barrier (ITB). Internal transport barrier were
first created in a magnetic-shear configuration in which
the generalization of the magnetic winding number q(r)
decreased as a function of radius in the center of the
tokamak. Between the center and the edge, q reached a
minimum and increased further out. This type of q
variation requires a current profile that peaks away from
the center of the plasma cross section, and therefore
tends to be transient because of inward current diffusion. In ITB’s, the turbulent diffusion of ion heat can
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
FIG. 12. Electron and ion thermal conductivities in a discharge
with an internal transport barrier. The ion conductivity falls
below a theoretical estimate for the collisional conductivity,
indicating that turbulent losses are extremely small. For reference, the shear straining rate ␻ E⫻B is compared with ␥ ␩ i , the
linear growth rate for the ion-temperature-gradient instability
with zero flow shear. The latter is an estimate for the turbulent
correlation rate. From Burrell, 1997.
diminish to the rate established by collisional processes.
This is illustrated in Fig. 12, which shows the experimental ion and electron thermal conductivities compared
with a standard theoretical prediction for the collisional
conductivity. The measured conductivity is actually below the predicted collisional conductivity in the central
region. (The standard theoretical expression used in
making Fig. 12 overestimates the true collisional conductivity because it was not formulated for the steep
gradients and strong electric field of the ITB and is not
valid when the distance to the center of the plasma is
comparable to a banana orbit width.) The comparison is
taken as an indication that the turbulence-driven conductivity is smaller than the collision-driven conductivity. With heat and particles continually supplied by neutral beams, the pressure gradient can become so steep
that global instabilities may set in. It is possible to tailor
the plasma parameters so that pressure profile steepening is distributed over the entire plasma, locally avoiding
pressure-gradient thresholds for instability, while
achieving transport reduction throughout the plasma.
The negative magnetic shear in the core (dq/dr
⬍1), in conjunction with q⬎1 throughout the plasma,
reduces the growth rate of key instabilities and suppresses coherent core fluctuations known as sawtooth
oscillations to an extent allowing a bifurcation of the
plasma equilibrium to a new state with strong flow
shear. The flow shear suppresses turbulence and transport as described in Sec. II. The transition or bifurcation
that produces the flow shear is necessary for enhanced
146
P. W. Terry: Suppression of turbulence by sheared flow
confinement, and there is a heating or power input
threshold required for the transition. However, the
negative magnetic-shear configuration, while helpful,
does not appear to be necessary (Koide et al., 1996; Burrell et al., 1998) if some other method is used for suppressing sawtooth oscillations. There is a temporal and
spatial correlation between an estimate of the suppression parameter ␧ s , Eq. (2.17), and the transport barrier.
In the experiments, ␧ s is not measured directly because
the turbulence correlation time is typically not available.
In its place, the instability growth time is used, calculated from kinetic linear stability codes in toroidal geometry that employ measured profiles for density, temperature, and current, but assume zero flow. Such codes
accurately reproduce the linear behavior of a large number of plasma collective instabilities. The replacement of
the turbulence correlation time with the linear growth
time is based on the approximate equality of these
times, which holds because turbulent energy transfer
saturates the instability. Although the equality is not exact, experience indicates that the approximation is often
quite good. Before the transition, the shear straining
rate and linear growth rate are comparable. Immediately after the transition, and throughout the enhanced
confinement mode, the shear straining rate exceeds the
linear growth rate. Likewise the shear straining rate exceeds the linear growth rate in the regions where density
fluctuations are suppressed and the transport fluxes are
reduced. Ion transport is strongly reduced in all devices
that have studied this enhanced-confinement mode,
while electron transport has been modestly reduced in
some cases and not significantly reduced in others. The
weaker effect on electron transport is not understood.
There are other enhanced-confinement modes that involve transport reduction in the core. In the high internal inductance discharges of DIII-D, characterized by a
transient highly peaked current profile, confinement improvement is correlated with an increased E⫻B flow
shear (Lao et al., 1993). The most intensively studied
enhanced-confinement mode in the Tokamak Fusion
Test Reactor (TFTR) was the supershot (Strachan et al.,
1987). This discharge was characterized by strongly
peaked pressure profiles in the core. The plasma was
heated with nearly zero-momentum input. From recent
analysis and modeling, Ernst et al., (1998) have argued
that the steep ion pressure gradient produces a significant radial electric field [see Eq. (4.1)] and associated
E⫻B shear flow. The flow shear is localized in the core,
yielding enhanced confinement in that region. Outside
the shear layer, confinement is degraded. Other
enhanced-confinement modes that steepen profiles, such
as the pellet enhanced-performance mode (Hugon et al.,
1992), may also produce a radial electric field in the region of a large pressure gradient (Carreras, 1997).
4. Externally induced internal transport barrier
Given the confinement improvements attained with
core transport barriers, particularly when produced in
combination with the H mode, a technique for externally inducing transport barriers is highly desirable. ExRev. Mod. Phys., Vol. 72, No. 1, January 2000
ternally induced barriers offer the possibility of controlling transport, e.g., placing the barrier at desired
locations to obtain turbulence-free operation, maintaining barriers in a steady state, or transiently weakening
barriers to reduce pressure gradients or exhaust the helium ash of fusion. The CH mode (LeBlanc et al., 1995)
discussed in Sec. IV. B. 3 represents an initial demonstration that externally induced transport barriers can be
formed using injected rf waves to produce an internal
flow shear layer. In the final experiments on the TFTR
tokamak at Princeton Plasma Physics Laboratory, the
formation of the CH mode was attempted on the larger,
hotter TFTR plasma using an rf injection scheme similar
to that of the PBX-M. The limited time available for the
experiment before shutdown of the device did not permit adjustments and optimization. A flow shear layer
was successfully created in the core, but the rf power
was insufficient to reach the suppression threshold, as
given by Eq. (2.17) (LeBlanc et al., 1999). This result is
encouraging and suggests that external transport control
may be possible in fusion-grade plasmas.
B. Stellarator
The stellarator and its variants (torsatron, heliotron)
are toroidal devices in which helical external current
windings produce both toroidal and poloidal field components. It is thus possible to operate a stellarator with
no plasma current, although in practice a plasma current
may exist. Because the helical windings break toroidal
symmetry, the radial electric field (as opposed to its radial derivative) has long been known to affect collisional
transport rates in stellarators (Mynick and Hitchon,
1983). Notwithstanding, the H mode has been observed
in the current-free Wendelstein 7-AS (W7-AS) stellarator (Erckmann et al., 1993) and in the CHS torsatron/
heliotron (Toi et al., 1994). The observation of the H
mode in stellarators is significant because it indicates
that the physics of the H mode is not dependent on the
high plasma currents of tokamaks and it suggests that
the radial derivative of the radial electric field is important in stellarators, as in tokamaks. The stellarator
evinces all the H-mode signatures of tokamaks. H ␣ and
D ␣ (alpha line of deuterium) emissions drop at the transition, fluctuations drop, gradients steepen, and the confinement time rises. The increase in confinement time is
weaker in a stellarator than it is in a tokamak. The maximal increase of 30% in W7-AS was attributed to the
presence of a limiter, which in tokamaks is known to
result in smaller confinement time increases. In W7-AS
the poloidal rotation rate of the BIV impurity species
also rises dramatically just preceding the H-mode transition. In a back transition to the L mode the poloidal
rotation rate returns to its lower L-mode value. Flow
shear, already sizable in the L mode, was not observed
to increase in the CHS. In the CHS, however, the confinement improvement was only 15%. In W7-AS, the H
mode occurs only for a narrow range of values of the
inverse winding number [the generalization for stellara-
P. W. Terry: Suppression of turbulence by sheared flow
tors of q ⫺1 from Eq. (5.3)]. For the observed range a
velocity shear layer already exists prior to the transition,
and neoclassical flow damping is minimum (Hirsch et al.,
1997). This narrow operational window is consistent
with a suppression of transport by flow shear. However,
the stellarator results overturn H-mode theories based
on specific current profiles or the large magnetic shear at
the edge of diverter tokamaks.
C. Reversed-field pinch
The reversed-field pinch (RFP) is a toroidal device in
which the toroidal and poloidal magnetic fields are comparable in magnitude (Taylor, 1986). The radial profile
of q decreases monotonically, starting at a few tens of
percent in the core and decreasing through zero to a
negative value of a few percent at the wall. In this configuration global tearing modes are unstable and dominate the core transport of heat and particles. The RFP
thus differs from the tokamak in two significant ways,
both of which have potential impact on suppression of
turbulence by flow shear. First, the turbulence in the
RFP is magnetic, whereas tokamak fluctuations are
thought to be dominantly electrostatic. Second, the turbulence in the RFP is global in scale, whereas in the
tokamak radial fluctuation scales are a few centimeters.
The latter distinction is particularly important since flow
shear in the H mode is confined to a narrow layer. These
differences have led to strategies for confinement enhancement in the RFP that have focused on reducing
the global magnetic turbulence by removing its freeenergy source, the current gradient. The injection of current into the edge to flatten the current gradient has led
to increases in confinement time by a factor of 4–5
(Sarff et al., 1997).
Recent observations indicate that a sheared E⫻B
flow is present in plasmas of the RFX experiment (Antoni et al., 1997) and the Madison Symmetric Torus
(MST; Chapman, Almagri, et al., 1998; Chapman,
Chiang, et al., 1998). In the MST, a shear flow occurs
with a pulsed current drive technique and is generated
spontaneously in other discharges labeled enhancedconfinement discharges. The contribution of flow shear
to confinement enhancement in the pulsed current drive
discharges has not been assessed precisely because the
current flattening effect is also present. In enhancedconfinement plasmas there is no current drive and the
confinement time increases by as much as a factor of 3
relative to a standard case with no flow shear and no
current drive. Both temperature and particle confinement times increase. Like the H mode, the flow shear is
confined to a narrow layer of approximately 1-cm width
in the edge. The presence of flow shear is correlated
temporally and spatially with a reduction in fluctuations.
The enhanced-confinement phase is initiated with a sawtooth crash, after which fluctuations of the magnetic
field and electrostatic potential decrease. The decrease
occurs over a broad frequency range that includes the
unstable global tearing modes. The spatial region of suppression extends throughout the edge, but suppression is
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
147
most pronounced in the region of flow shear. Some
steepening of the temperature and density gradients in
the edge is observed or inferred. These observations
complement the studies of confinement changes in the
reversed-field pinch with biasing (Craig et al., 1997).
The observation of turbulence suppression with flow
shear in the RFP raises issues that have not been confronted in tokamak research, including the effect of a
narrow shear layer on global fluctuations and the generation of flow by a magnetic Reynolds stress of turbulence that is predominantly nonpropagating. Initial work
on the effect of a narrow flow shear layer in the external
region of a tearing mode indicates that localized external flow shear can suppress a global tearing mode, provided it is sufficiently strong (Hegna et al., 1998). However, there is also electrostatic edge turbulence in the
RFP, and its suppression may lower the edge resistivity,
indirectly diminishing tearing-mode instability. The
magnetic Reynolds stress is typically zero in the RFP, as
expected for global tearing modes, but becomes very
large in sawtooth events (Fiksel, 1998). The latter are
precursors for the enhanced-confinement phase. Flow in
the RFP is mostly toroidal, and, unlike the tokamak,
there is no significant neoclassical viscosity. These properties suggest flow generation and transition processes
that differ in crucial ways from those of other devices.
D. Linear magnetic configurations
Linear devices represent the greatest variation of geometry from that of the tokamak, where flow-shearinduced transport barriers were first observed. In linear
devices, magnetic-field lines may terminate on material
surfaces, e.g., the end plates of a cylindrical vacuum vessel, and plasma can be lost by axial motion.
The tandem mirror is a cylindrical device with an axial
magnetic field whose strength increases toward the cylinder ends. Particles are trapped in the magnetic mirror
thus formed, but only those with velocities along the
field that are not sufficient to overcome the mirror barrier. End cells with electrostatic potentials inhibit the
loss of particles that escape the magnetic mirror. The H
mode has been achieved on a tandem mirror by biasing
an annular limiter placed midway along the cylinder axis
(Sakai, Yasaka, and Itatani, 1993). The transport barrier
so formed arrests radial losses, but not axial losses. In
the H mode, neutral emissions abruptly decrease, the
bulk density and energy rise, fluctuations decrease, and
gradients steepen in the region of E⫻B flow shear. The
confinement time increases by a factor of 2.5. The
voltage-current characteristic of the limiter exhibits a bifurcation like that of tokamak biasing experiments, with
a sharp drop in current for a given voltage just above the
bifurcation threshold. Also, as observed in other experiments, the reduction of transport begins with the flow
shear created before the bifurcation, indicating that the
transport reduction is intrinsic to the flow shear and not
the bifurcation. The standard interpretation of the bifurcation in tokamaks involves the nonlinear dependence
of the neoclassical viscosity on flow speed. Because the
148
P. W. Terry: Suppression of turbulence by sheared flow
neoclassical viscosity is intrinsic to toroidal geometry
(Sec. IV. B), the mechanics of the bifurcation in mirrors
remains to be understood.
Another linear device is the Z pinch. The Z pinch is
cylindrical, but the magnetic field is azimuthal, created
by the axial plasma current (Freidberg, 1982). The
Z-pinch configuration represents a solution of the ideal
MHD equilibrium model, either for a steady state or
under implosion by the radial Lorentz force produced
by a large axial current. The Z pinch is susceptible to a
number of rapidly growing Alfvén-time-scale instabilities. Following up on experimental indications of improved performance in the presence of flow, Shumlak
and Hartman (1995) found that a sheared axial flow stabilizes the m⫽1 kink mode, provided the equilibrium is
stable to the m⫽0 mode and the flow shear strength
dU z /dr exceeds 0.1k v A . (U z is the axial equilibrium
flow and k and m are the axial and azimuthal mode
numbers.) If the pressure profile does not satisfy the
precondition for m⫽0 stability, stabilization of ideal
MHD modes requires supersonic flows as large as Mach
4 (Arber and Howell, 1996a). It has also been argued
that flow shear stabilizes the Rayleigh-Taylor instability
in imploding Z-pinch plasmas (Shumlak and Roderick,
1998). Given the existence of certain numerical issues
(Arber and Howell, 1996b; Shumlak and Hartman,
1996) and the indirect connection between infinitesimalamplitude linear stability analyses and experimental
conditions, further experimental work is needed to ascertain whether flow shear can provide the benefits for
the Z pinch that exist with the tokamak.
VII. TRANSITION TO ENHANCED-CONFINEMENT STATES
A. Bifurcation of plasma state
Sections II and IV effectively treated as independent
the suppression of turbulence by flow shear and the generation of flow shear by a variety of mechanisms, including turbulence. In reality, the two processes are coupled
and must be treated self-consistently. For illustration,
consider a closed set of equations, taken from prior sections, that describes some of the couplings:
⳵␰
⳵␰
⳵␰
⳵␰
⫹u E 共 x 兲 ⫹ũ xi ⫹ũ yi ⫽ ␴ ␰ ,
⳵t
⳵y
⳵x
⳵y
(7.1)
⳵
Er
1
p ⫺ 具 u yi 典 ,
⫽
B ␸ B ␸ Z i en i ⳵ x i
(7.2)
u E共 x 兲 ⫽
⳵ 具 u yi 典
⳵
⫽⫺
具 ũ ũ 典 ⫺ ␮ ␪ 具 u yi 典 ,
⳵t
⳵ x xi yi
(7.3)
⳵pi
⳵
⫹
Q ⫽P̂ i .
⳵t
⳵x i
(7.4)
Equation (7.1) is the equation for turbulent fluctuations
[Eq. (2.12)], and in this system includes both ion flow ũi
and ion pressure p i . Each fluctuation has a source ␴ ␰ ,
and the shear flow u E (x) is a poloidal E⫻B flow with
shear in the radial (x) direction. The E⫻B flow is deterRev. Mod. Phys., Vol. 72, No. 1, January 2000
mined by Eq. (7.2) [Eq. (4.1) divided by the magnetic
field]. The magnetic field is taken as toroidal. For simplicity, there is no mean toroidal flow. The poloidal ion
flow is governed by Eq. (7.3) [Eq. (4.4) without magnetic
fluctuations]. The mean pressure p i is governed by Eq.
(7.4) [Eq. (2.34)], where the ion heat flux Q i is the correlated product of fluctuations of the ion flow and pressure, Q i ⫽ 具 ũ xi p̃ i 典 , and P̂ i is the external input power
density to the ions.
In systems of equations such as Eqs. (7.1)–(7.4), quantities feed back on themselves through the couplings.
For example, (1) from Eqs. (7.2), (7.1), and (7.4), ⵜp i
drives u E (x), u E (x) suppresses ũi and p̃ i , lower values
of ũi and p̃ i reduce Q i , reduced Q i increases ⵜp i , which
further drives u E (x); (2) when poloidal flow dominates
the pressure force equation (7.2), 具 ũ xi ũ yi 典 drives 具 u yi 典 ,
which drives u E (x), and through Eq. (7.1), u E (x), determines the value of ⳵ / ⳵ x 具 ũ xi ũ yi 典 . The feedbacks of
Eqs. (7.1)–(7.4) are nonlinear and spatiotemporal and
give rise to the striking phenomenon of dynamic transitions between different mean states of the system. These
transitions are generally referred to as bifurcations because small changes in an externally controlled parameter lead to abrupt and large changes in the character of
the plasma state.
Bifurcations have been observed in the external biasdriven H mode in tokamaks, stellarators, and mirrors,
where the bifurcation is driven by increasing the bias
voltage and is evident in both the I-V curve of the electrode and the plasma equilibrium. In a reversed-field
pinch, biasing leads to improved confinement, but a bifurcation has not been observed. A bifurcation occurs in
the spontaneous H mode, in the spontaneous enhancedconfinement mode of the RFP, in the internal transport
barrier, and in the core barrier in JT-60U. These bifurcations are induced by increasing the input heating
power. From the above statement it is evident that bifurcated states entail fluctuation suppression, but suppression is not always accompanied by bifurcation.
Backward transitions can also be induced by reversing
the direction of changes in the external parameter that
controls the transition. Hysteresis is generally evident,
i.e., the threshold values of the control parameter are
different for forward and backward transitions.
This section describes aspects of the phenomenology
and modeling of transitions in order to illustrate the rich
nonlinear dynamics that underlie transport barriers in
fusion plasmas. The current state of transition modeling
is primitive and heuristic. At present, no transition
theory captures every relevant detail of experimental
transitions, making it difficult to assess their validity.
Nevertheless, a variety of transition models have been
disproved by experiment, or their dominant processes
found to be of marginal importance. Other models, including some discussed here, have known shortcomings,
but may include physics of importance in the actual transition.
B. Transition modeling
Analytic transition models can be separated into twostep and single-step models. In two-step models, flow is
P. W. Terry: Suppression of turbulence by sheared flow
first driven through a bifurcation that relies on the nonlinear variation of the neoclassical poloidal flow damping with flow speed. In the second step there is suppression of turbulence, profile steepening, and increased
confinement. These result from the increased flow speed
that accompanies the bifurcation, but do not contribute
causally to the bifurcation. In single-step models, flow
generation and turbulence suppression evolve as integral parts of the transition and cannot be separated.
Nonlinear poloidal flow damping is not required. Singlestep transitions are divided into first- and second-order
critical transitions. Transitions have recently been reported in the direct numerical simulation of edge fluid
models (Rogers, Drake, and Zeiler, 1998). It has not
been ascertained how these results compare with theoretical models.
1. Two-step transition models
When a radial electric field and E⫻B flow are externally driven, as in biased electrode experiments, the
transition is intrinsically a two-step process (Cornelis
et al., 1994). A two-step scenario has also been proposed
for the H-mode and internal barrier transitions, invoking a plasma drive mechanism for flow shear generation
(Shaing and Crume, 1989; Shaing et al., 1998). Because
these models rely on nonlinear poloidal flow damping
for bifurcation, they apply only to situations in which the
dissipation is dominated by poloidal flow dynamics.
The transition observed in biased electrode experiments was modeled with the ion momentum and
electric-field equations [Eqs. (4.3) and (4.1)] using experimental data to supply equilibrium quantities, such as
the pressure gradient, at any desired time during the
transition (Cornelis et al., 1994). These two equations
combine to yield a radial Ohm’s law with an effective
radial conductivity that depends on neoclassical flow
damping. The Ohm’s law is obtained by first noting that
an equilibrium plasma current flows in the radial direction in response to a current injected into the plasma
from the electrode. The plasma current enters the ion
momentum balance [Eq. (4.3)] through the J⫻B force.
The other contributors to the balance are the neoclassical viscous damping and ion-neutral friction, yielding
J
1
ⵜ•⌸
⫹m i ␷ i0 ui .
共 J r r̂⫻B兲 ⫽
ni
ni
(7.5)
J
where n ⫺1
i ⵜ•⌸ can be cast as the neoclassical viscous
force m i ␮ ␪ u ␪ i and ␷ io is the ion-neutral damping rate.
This equation can be solved for ui and substituted into
the equation for the radial electric field [Eq. (4.1)] to
yield Ohm’s law. The conductivity is equal to a linear
combination of the neoclassical flow-damping rate and
the ion-neutral collision rate. As noted in Sec. IV. B. 2,
the neoclassical flow-damping rate ␮ ␪ has a maximum
value at u ␪ ⫽u Ti B ␪ /B ␸ (see the dotted curve in Fig. 1 of
Shaing and Crume, 1989). Above this critical flow, which
corresponds to the critical electrode current and bias
voltage, the neoclassical viscosity drops, leading to a
larger flow and smaller electrode current. Cornelis et al.
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
149
(1994) compare measured and predicted radial electricfield profiles, using a variety of theoretical models for
the neoclassical viscosity. All agree qualitatively with
the experimental data, but there are distinct quantitative
differences.
The two-step model for a spontaneous H-mode transition invokes ion-orbit loss to drive a radial electric field
within a banana orbit of the separatrix (Sec. IV. B.4).
Through orbit squeezing, the width of the banana orbit
depends on the E⫻B flow shear, becoming smaller as
the shear increases. If, through a similarity relation, the
flow shear is proportional to the flow, the torque supplied by orbit loss is maximum for zero flow and falls to
zero at higher flow speeds due to the squeezing effect.
4
The torque goes as ␷ i ( ␷ i ⫹ ␣ 4 u pm
) ⫺1/2 exp关⫺(v i
* *
*
4 4 1/2
⫹␣ upm) 兴 , where v i is the rate at which ions are col*
lisionally scattered across the separatrix from a trapped
orbit to an untrapped orbit, ␣ is the form factor that
relates the flow shear to the flow speed (Shaing and
Crume,
1989),
and
u pm ⫽(u Ti B ␪ ) ⫺1 关 u ␪ B ␸
⫺1
⫺(Z i en i ) ⵜp i 兴 is the poloidal flow, normalized and
adjusted by the pressure contribution in the radial momentum balance equation. The collision rate ␷ i is nor*
malized to the bounce rate, or inverse time for an ion to
move between the turning points in the magnetic mirror.
For an ion to be trapped in the mirror field, and therefore have a banana orbit, ␷ i must be less than unity,
*
i.e., the ion must not scatter from its trapped orbit before reaching a turning point. The neoclassical viscous
force is linear at small flow velocities, where it behaves
like a standard viscous drag. Above u E ⫽u Ti B ␪ /B ␸ ⬇1,
it turns over and falls to zero. The viscosity also has an
overall proportionality to the collision rate v i , because
*
the drag involves collisional friction between untrapped
and trapped ions.
The steady-state E⫻B flow is determined from the
balance of the applied torque and the viscous force.
When the collision rate ␷ i is large, the torque and the
*
drag balance at a small flow speed, in the parameter
region where the viscous force is linear in flow speed.
This balance corresponds to the L mode. As the collision rate decreases, three solutions become possible.
One continues the branch with small flow speed and two
occur for a larger flow speed, in the parameter region
where the viscous force becomes weaker with increasing
flow speed. The branch with the largest flow speed is
stable (as is the L-mode branch) and represents the H
mode. At still weaker collision rates, the H-mode branch
is the only solution yielding a steady flow. The appearance of the H-mode branch allows for a bifurcation of
the equilibrium to a state with large flow shear.
In this transition model, the transport barrier is
formed within approximately one poloidal gyroradius
(ion banana width) [Eq. (4.6)] of the separatrix or limiter. The width of the barrier region is squeezed by E
⫻B flow, so the poloidal gyroradius is an upper limit.
For electric-field generation, ␷ i must be less than unity,
*
and the poloidal Mach number should change from less
than one to greater than one as the plasma goes from L
mode to H mode. Experimental data on the H-mode
150
P. W. Terry: Suppression of turbulence by sheared flow
transition have been examined for evidence of these features. While the model reproduces qualitative features
of the H mode, there are serious quantitative discrepancies related to these theoretical constraints. Measurement of u pm before and after the transition shows that it
passes through the critical value of unity, consistent with
theory (Burrell et al., 1992). However, the barrier width
does not change under variation of the poloidal field and
can be as large as six times the poloidal gyroradius. The
normalized collision rate ␷ i is more than an order of
*
magnitude larger than unity in JFT-2M (Ida et al., 1990)
and can vary from 1 to 14 in DIII-D (Carlstrom and
Groebner, 1996).
A two-step transition theory has also been proposed
for the transition to the internal transport barrier in
TFTR (Shaing et al., 1998). The ion-electron flux difference created by the large toroidal ripple of TFTR (Sec.
IV.C) is invoked to create a radial electric field in the
core. Although the plasma rotation is principally toroidal, the dominant dissipation process is the damping of
poloidal flow. Like two-step edge bifurcation models,
the nonlinear variation with flow of the neoclassical poloidal viscosity produces the bifurcation. The internal
transport barrier transition in TFTR produces a rapid
poloidal spinup with a subsequent return to pretransition rotation rates on a slightly longer time scale (Bell
et al., 1998). (As the flow relaxes, the pressure gradient,
steepened by suppression of the heat flux, takes over as
the driving mechanism for the radial field.) Because twostep transitions are governed by the nonlinearity of the
poloidal flow damping, the decay of the flow transient is
predicted to occur over a collisional relaxation time,
which is much longer than that observed in TFTR.
2. First-order critical transition theory
In first- and second-order critical transition models,
the suppression of turbulence by E⫻B flow shear is an
integral part of the transition dynamics. In analogy with
the Landau theory of phase transitions (Landau and Lifshitz, 1980), the flow shear is the order parameter. In the
simplest first-order model (Hinton, 1991), equilibrium
balances from Eqs. (4.1) and (4.3) are combined with a
model for the heat flux that incorporates shear suppression heuristically. The radial electric field is governed by
Eq. (4.1) with the poloidal flow term dominating the
right-hand side, E r ⫽⫺u ␪ B ␸ . The poloidal flow is governed by Eq. (4.3) from a balance of terms that make up
J . Using the
the anisotropic pressure force (n i ) ⫺1 ⵜ•⌸
J
standard neoclassical evaluation of ⵜ•⌸ (Hinton and
Hazeltine, 1976), we find that the flow is proportional to
the ion temperature gradient, u ␪ ⫽⫺(eB) ⫺1 ␮ ⳵ T/ ⳵ r,
where the dimensionless constant of proportionality ␮ is
given by ␮ ⫽1.7⫹ ␮ 1 ( ␷ i ) ⫺2 and ␮ 1 is a constant less
*
than unity. This model does not invoke nonlinear variation of the flow damping above the poloidal sound
speed. The radial shear in u ␪ , which directly relates to
the E⫻B flow shear, is given by
冉 冊
⳵u␪
⳵T
4␮l
⫽⫺
2
⳵r
eB 共 ␷ i 兲 T ⳵ r
*
2
,
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
(7.6)
where the additional temperature-gradient factor arises
from the dependence of ␷ i on the temperature, and the
*
second derivative of T is assumed to be smaller than the
radial variation of ␷ i that enters through the tempera*
ture gradient.
The local heat flux [Eq. (2.9)] is modeled by the expression
Q⫽Q n ⫹
Qf
,
1⫹ ␥ f 共 ⳵ u ␪ / ⳵ r 兲 2
(7.7)
where Q n is the neoclassical (collisional) heat flux, Q f is
the fluctuation-induced heat flux in the absence of flow
shear effects, and the denominator is a model representation of shear suppression with ␥ f a constant. For illustration, assume a Fick’s law variation for the fluxes Q n
and Q f , making both linear functions of the temperature gradient. The fluctuation-induced heat flux dominates the collisional heat flux when there is no shear
flow, so the linear rise of Q f with the temperature gradient is steeper than that of Q n . The shear suppression
factor introduces additional dependence on the temperature gradient through Eq. (7.6), yielding
Q⫽Q n ⫹
Qf
,
1⫹␭ f 共 ⳵ T/ ⳵ r 兲 4
(7.8)
where ␭ f ⫽ ␥ f (4 ␮ 1 /eBT) 2 ( ␷ i ) ⫺4 . For reasonable val*
ues of the parameters ␭ f and Q n /Q f , the net variation
of the total local heat flux is as follows. For ⳵ u ␪ / ⳵ r
⬍␭ f⫺1/4 , the local heat flux is dominated by the
fluctuation-induced flux with negligible shear suppression, Q⬵Q f . For gradients just larger than ␭ f⫺1/4 , the
total local heat flux decreases with increasing gradient as
a result of shear suppression. At still larger gradients the
suppression of the fluctuation-induced flux is so strong
that the local heat flux goes as the collisional flux, Q
⬵Q n . This behavior is shown in Fig. 13.
The transition is manifested as a discontinuity of the
temperature gradient. The discontinuity appears when
the temperature gradient is determined by equating the
local heat flux [Eq. (7.8)] to the heat flux at the boundary of the local region under consideration. When the
auxiliary heating power is low, the boundary flux is low.
Equating Eq. (7.8) with a small boundary flux yields a
temperature gradient from the linear part of Eq. (7.8)
where Q⬵Q f . The boundary flux has a critical value
when it is equal to the local heat flux at its maximum,
where flow shear begins to suppress the fluctuationinduced heat flux. For a boundary flux slightly above the
critical value, the intersection shifts discontinuously to
the part of Eq. (7.8) where Q⬵Q n , and the gradient is
much larger than g 1 . The flow shear also changes discontinuously as the boundary heat flux passes through
the critical value. The temperature profile is determined
by equating Q with the boundary heat flux at each point
and integrating from the edge inward. If the boundary
flux at the edge is above the critical value, the boundary
flux deeper in the plasma falls below the critical value,
P. W. Terry: Suppression of turbulence by sheared flow
151
fluctuation strength in the absence of flow shear has a
spatial variation to account for the stabilizing effect of
magnetic shear. The shear suppression factor is equivalent to that of Eq. (7.7). The poloidal flow is governed
by the balance of pressure force and flow damping as in
Hinton (1991), but the density profile is evolved instead
of the temperature profile. This leads to an equation like
Eq. (7.6), with ⳵ n/ ⳵ r replacing ⳵ T/ ⳵ r. The transition begins in the core where the fluctuation energy is minimum
because of magnetic-shear stabilization and works its
way outward until it stops at a position where the flow
damping lowers the flow shear below the suppression
threshold.
FIG. 13. Local heat flux as a function of the temperature gradient in a model for first-order critical transitions. For small
values of the gradient the flux is dominated by turbulence
(Q⫽Q f ). For larger gradients, flow shear suppresses turbulent
heat transport and approaches the collisional heat flux Q n
asymptotically. The temperature gradient is determined by
equating the local flux [Eq. (7.8)] with a boundary flux at the
edge of the local region. When the boundary flux just exceeds
Q crit , the gradient shifts for g 1 to a value near g 2 . Adapted
from Hinton, 1991.
and the gradients inside that point correspond to the
part of the local flux curve where Q⬵Q f and ⳵ T/ ⳵ r
⬍g 1 . A steep temperature gradient and strong flow
shear thus form in an edge region, as in the H mode.
This transition is analogous to the first-order critical
transition occurring, for example, between the liquid
and gas phases of a fluid (Landau and Lifshitz, 1980).
The order parameter can be chosen to be either the temperature gradient or the flow shear. The temperature
parameter of the Landau phase transition is replaced by
the boundary heat flux. As the boundary flux passes
through the critical value, the order parameter jumps
discontinuously. Figure 14 gives a schematic representation of this behavior. The jump in flow shear is analogous to the presence of latent heat in the liquid/gas
phase transition. Characteristic of first-order phase transitions, there is hysteresis (Hinton, 1991). When the
boundary heat flux is reduced from a large value
through the critical point (H-L transition), the stable
equilibrium on the Q⬵Q n part of Eq. (7.7) remains
stable down to the minimum of Q. For lower values, the
gradient shifts discontinuously to smaller values where
Q⬵Q f . The critical flux for the back transition is thus
lower than the critical flux for the forward transition.
An extension of the Hinton (1991) model incorporates both the particle and heat flux, yielding profiles for
both density and temperature (Hinton and Staebler,
1993). Time-dependent effects associated with inward
motion of the critical point in the L-H and VH transitions have also been modeled (Staebler et al., 1994). A
first-order transition model has been developed for the
internal transport barrier transition (Diamond et al.,
1997). The local heat flux is replaced by an equation for
the fluctuation energy. (These models do not account for
the cross-phase effects of Sec. V.C, so the heat flux is
simply proportional to the fluctuation energy.) The local
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
3. Second-order critical transition theory
A second-order critical transition is possible when the
Reynolds stress is larger than the ion pressure term in
the ion momentum balance (Diamond et al., 1994). The
Reynolds stress drives a sheared poloidal flow against
the dissipation of neoclassical flow damping. The
sheared poloidal flow produces E⫻B flow shear, which
reduces fluctuations. The magnitude of the Reynolds
stress depends on both the fluctuation energy and the
flow shear. Two basic equilibrium states are possible under this coupling. There is a state in which the Reynolds
stress is too weak to overcome the flow damping. Any
perturbation of the flow decays to zero. The fluctuation
energy is large because there is no flow shear. In the
other state, the Reynolds stress is sufficient to drive a
steady flow, whose shear suppresses turbulence. A transition to the second state represents a diversion of free
energy from fluctuations into the mean flow, through the
Reynolds stress.
A dynamical model for the second-order transition is
derived from the poloidal momentum equation, Eq.
(4.4), and an equation for the turbulent energy evolution. The latter is constructed from an appropriate turbulent amplitude equation by multiplying by the amplitude and taking an average over a suitable statistical
ensemble. Equation (7.1) is a generic amplitude equation and Eq. (5.10) is a specific example for drift-wave
turbulence. A statistical closure is applied so that the
nonlinearity is proportional to the square of the turbulent energy. This process is detailed for Eq. (5.10) in
Carreras et al. (1992). The form of the energy equation
is
1 ⳵E
⫽ ␥ 0 E⫺ ␣ 1 E 2 ⫺ ␣ 2 UE,
2 ⳵t
(7.9)
where E is the fluctuation energy of the linearly unstable, energy-containing modes, and U is the square of
a uniform mean flow shear strength in the region of the
unstable modes. The first term on the right-hand side is
the linear instability term, describing injection of energy
at the rate ␥ 0 . The second term represents spectral
transfer of energy away from the unstable modes into
damped modes at the rate ␣ 1 E. In a steady state without flow shear, the turbulent energy is fixed by the balance of the injection rate with the spectral transfer rate,
152
P. W. Terry: Suppression of turbulence by sheared flow
yielding E⫽ ␥ 0 / ␣ 1 . The third term describes shear suppression, lowering the steady-state turbulent energy
when U is nonzero. Specific forms for ␥ 0 , ␣ 1 , and ␣ 2 are
given in Diamond et al. (1994) for a variety of turbulence models.
An equation for U is obtained by taking a radial derivative of the poloidal ion momentum balance and multiplying by the radial derivative of the mean poloidal
flow. This equation must also be closed in order to express the Reynolds stress as a function of U and E. The
equation for U is
1 ⳵U
⫽⫺ ␮ U⫹ ␣ 3 UE,
2 ⳵t
(7.10)
where ␮ is the poloidal flow-damping rate and ␣ 3 is a
constant. The first term on the right-hand side is the
neoclassical flow-damping rate, modeled in the linear regime below the poloidal sound speed, and the second
term is the closure of the Reynolds stress. The definition
of the Reynolds stress [Eq. (2.11)] directly leads to linear scaling with fluctuation energy. The linear dependence on U reflects the fact that the asymmetry required
for a nonzero Reynolds stress can be induced by the
flow shear (as it shears and tilts fluctuations). For example, consider the Reynolds stress of the linear eigenmode of the drift-wave equation (5.10). The eigenmode
is (Carreras et al., 1992)
␾ k ⫽ ␾ 0 exp
冋
⫻exp
冉
1⫹i ␰ k2
&
W k2
⫺i
2&W k2
冊 冋
exp
⫺1
2&W k2
共 x⫺ ␰ k 兲 2
册
共 x⫹ ␰ k 兲 2 ,
册
(7.11)
where W k is the mode width ⌬ evaluated at the oscillation frequency ␻ ⫽k y U(0)⫹k y U D , and ␰ k ⫽⌬ 3 S/2␳ s2 is
proportional to the flow shear. The other parameters
were defined in Sec. V.A.2. When the flow shear is zero,
␰ k ⫽0, and the radially symmetric Gaussian gives a zero
Reynolds stress. The asymmetry induced by ␰ k leads to a
nonzero Reynolds stress, whose radial derivative is
(Terry et al., 1994)
⳵
⳵
具 ũ ũ 典 ⫽C s2 ␳ s2
⳵x x y
⳵x
⫽C s2 ␳ s2
兺
k
y
冓
⫺ik y ␾ ⫺k
k y ␰ k ⌬ ⫺3 兩 ␾ 0 兩 2 ,
兺
k
⳵␾k
⳵x
冔
(7.12)
y
where ␾ k and ␾ 0 are dimensionless potentials under the
normalization by T e / 兩 e 兩 introduced with Eq. (5.10). The
factor ␰ k leads to the linear scaling with respect to U of
the third term of Eq. (7.10). Equation (7.12) is a manifestation of the radial symmetry breaking required for a
nonzero Reynolds stress, as discussed in Sec. IV.B.1.
Equations (7.9) and (7.10) have fixed points given by
(1) E⫽ ␥ 0 / ␣ 1 , U⫽0 and (2) E⫽ ␮ / ␣ 3 , U⫽( ␥ 0
⫺ ␣ 1 ␮ / ␣ 3 )/ ␣ 2 . The first fixed point corresponds to an
L-mode state with no flow shear and a fluctuation energy given by the balance of linear growth rate and specRev. Mod. Phys., Vol. 72, No. 1, January 2000
FIG. 14. First-order critical transition with dU ␪ /dr as the critical parameter and Q boundary as the effective temperature in the
Landau theory. The order parameter is discontinuous at Q crit .
tral transfer rate. It is stable when ␥ 0 ⬍ ␣ 1 ␮ / ␣ 3 . The second fixed point has finite flow shear and a suppressed
fluctuation energy and thus corresponds to the H mode.
It is stable for ␥ 0 ⬎ ␣ 1 ␮ / ␣ 3 . The control parameter can
be taken as the linear growth rate ␥ 0 , which increases
when auxiliary heating is applied and gradients are
steepened by the deposited power. (It could also be
taken as the damping rate ␮, which decreases when auxiliary heating is applied and the edge temperature is increased.) When the control parameter ␥ 0 increases, the
L-mode fixed point becomes unstable and the system
transits to the stable H-mode fixed point. The flow shear
increases from zero to its H-mode value as the fluctuation energy drops from its L-mode value to the reduced
H-mode value.
The flow shear is continuous across the transition, as
can be seen by evaluating the H-mode flow U⫽( ␥ 0
⫺ ␣ 1 ␮ / ␣ 3 )/ ␣ 2 at the transition threshold ␥ 0 ⫽ ␣ 1 ␮ / ␣ 3 .
The derivative of U with respect to ␥ 0 is discontinuous.
This behavior is analogous to second-order critical transitions in the Landau theory. The order parameter can
be chosen to be U 1/2⬇du ␪ /dr. The growth rate is the
effective temperature parameter. The transition, illustrated in Fig. 15, can be directly compared to the firstorder transition of Fig. 14, as both have the same order
parameter, and the temperature parameters are directly
related through the dependence of the turbulent heat
flux on the instability drive ␥ 0 . As a second-order transition, there is no hysteresis in the transition threshold.
However, the rate at which the system transits from one
fixed point to the other is different in the forward and
backward transitions (see Fig. 2 of Diamond et al.,
1994).
Second-order transition dynamics have been invoked
to explain the transient poloidal spinup in the internal
transport barrier transition (Newman et al., 1999). The
Reynolds stress generates a flow that initiates shear suppression and the steepening of the pressure gradient. As
the pressure gradient increases, it becomes the primary
driver of the radial electric field and the flow relaxes to
its pretransition state. Both the spinup and relaxation
P. W. Terry: Suppression of turbulence by sheared flow
153
burning plasma. Under some circumstances ELM’s behave as a transient return to L mode. This has led to
modeling studies of ELM’s based on limit-cycle solutions of transition models like the ones presented in this
section. It is also possible that ELM’s are a global instability driven by the steep gradients caused by transport
reduction. While ELM’s are an important issue in the
use of transport barriers in fusion plasmas, they are beyond the scope of this review. A discussion of ELM’s
can be found in Itoh and Itoh (1996).
VIII. GENERALIZATION TO RELATED SYSTEMS
FIG. 15. Second-order phase transition associated with shear
generated by the Reynolds stress. There is no discontinuity of
the order parameter dU ␪ /dr⬇U 1/2, but its slope is discontinuous at the critical value of the effective temperature
parameter.
are driven by the Reynolds stress and occur on similar
time scales, as observed in experiment. A transient spike
in the poloidal flow also occurs in models that include
stabilizing flow curvature as the dominant agent for turbulence suppression (see Fig. 2 in Terry et al., 1994).
Both cases illustrate a reciprocity principle intrinsic to
Reynolds stress-driven transitions. Whenever there is
sufficient shear to suppress fluctuations, the Reynolds
stress as modified by the shear is of sufficient magnitude
to modify the flow shear. Even in first-order transitions,
this holds for a few turbulent correlation times before
reduced fluxes have steepened the pressure gradient to
the point where it can dominate the radial force balance.
Thus second-order transition dynamics are likely to apply on short time scales, with first-order transition dynamics entering on longer time scales of the pressure
evolution. Second-order transition dynamics also play a
role in bifurcation models based on Galerkin projections
for quasicoherent nonlinear evolution close to a stability
threshold (Beyer and Spatschek, 1996), complementing
treatments that assume fully developed turbulence.
Second-order transitions have elements in common with
temporally and spatially evolving Reynolds stress-driven
atmospheric flows such as the quasibiennial oscillation
(Holton and Lindzen, 1972; Plumb, 1977).
C. Transient phenomena
Fluctuation activity does not cease after transition. In
addition to residual small-scale turbulence, the H mode
has a class of fluctuations known as edge-localized
modes (ELM’s). There are several types of ELM’s distinguished by gross features, including amplitude, frequency, and temporal characteristics relating to intermittent behavior. Large-amplitude ELM’s can eject a
fraction of the particles and heat stored in the plasma
edge region without terminating the plasma discharge.
They thus provide some control of edge gradients. In the
future they may be useful in removing helium ash from a
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
Previous sections have presented extensive experimental, analytical, and computational work describing
the suppression of turbulence and transport by E⫻B
flow shear in fusion plasmas. As is apparent from the
generality of Sec. II, shear suppression is a universal feature of advection of turbulence by stable, sheared mean
flow. In this section generalizations and applications to
nonplasma systems are considered. As in fusion plasmas,
turbulence in these systems is approximately 2D, and
the generalization carries over directly. The effect of
stable flow shear on turbulent decorrelation in fully 3D
systems remains to be investigated.
A. Coherent vortices in navier-stokes turbulence
Suppression of turbulent vorticity transport by flow
shear leads to spatial intermittency in decaying 2D
Navier-Stokes turbulence (Terry, 1989; Terry, Newman,
and Mattor, 1992). Intermittency manifests itself as the
emergence of coherent vortices in simulations that initialize homogeneous turbulence from a Gaussian random distribution of vorticity with no mean flow (McWilliams, 1984). As the turbulence decays under a
hyperviscous damping, certain eddies emerge as coherent vortices, avoiding mixing by ambient fluctuations
and persisting for a large number of eddy turnover
times. The vortices are patches of intense localized vorticity characterized by a particular variation within the
vortex of a quantity called the Gaussian curvature. The
Gaussian curvature is the difference of the mean
squared shear stress, ( ⳵ V/ ⳵ x⫺ ⳵ U/ ⳵ y) 2 ⫹( ⳵ U/ ⳵ x
⫹ ⳵ V/ ⳵ y) 2 , and the mean squared vorticity. (Here U and
V are the total flow velocities in the x and y directions.)
The Gaussian curvature is strongly positive in the vortex
core and strongly negative in the edge. Contours of
Gaussian curvature for a coherent vortex are shown in
Fig. 16. Because the coherent vortices avoid transferring
their vorticity into the turbulent cascade, they persist
long after ambient turbulence has been viscously dissipated through its cascade to the Kolmogorov scale (Sec.
II.A.1). A central question in the physics of intermittency is what physical process or processes permit a coherent vortex to avoid the turbulent mixing that causes
eddies to decay in an eddy turnover time. One such process is shear suppression of turbulence. Localized vorticity fluctuations have a flow profile in which flow shear is
largest at their edges. Those whose initial vorticity is
154
P. W. Terry: Suppression of turbulence by sheared flow
sufficiently more intense than ambient fluctuations have
an edge flow shear that satisfies Eq. (2.17), and ambient
turbulence and its transport of vorticity are suppressed.
Those whose initial vorticity is comparable to that of
ambient turbulence cannot suppress ambient turbulence. They participate in the cascade of energy to the
Kolmogorov scale and decay in an eddy turnover time.
The interaction of an intense symmetric vortex with
the ambient turbulence can be described by a two-time
scale analysis of the Navier-Stokes equation. The origin
of a polar coordinate system is placed at the center of
the vortex. With a Fourier-Laplace transform of the turbulent vorticity ␰ (r, ␪ ,t),
␰ n, ␥ ⫽
冕
⬁
0
dt exp共 ⫺ ␥ t 兲
冕
2␲
0
d ␽ exp共 in ␽ 兲 ␰ 共 r, ␽ ,t 兲 .
(8.1)
the n⫽0 component is the symmetric vortex and n⭓1 is
the turbulence. Turbulent vorticity fluctuations will be
referred to as eddies. The n⫽0 component evolves on a
slow time scale under the action of turbulent mixing. On
a rapid time scale the vortex can be treated as stationary. The evolution of the turbulence is identical to that
of Sec. II.B.3 and is given by Eq. (2.29),
关 ␥ n ⫺i 共 r⫺r 0 兲 n 共 ū/r 兲 r 兴 ␰ n, ␥ ⫺
⫹
冉
⳵␰ n, ␥
1⳵
rD n
r⳵r
⳵r
冊
⳵⌶
n2
⫺in
,
D ␰ ⫽
␾ n, ␥
r 2 n n, ␥
r
⳵r
(8.2)
where the terms on the left-hand side are defined after
Eq. (2.29), the source is the turbulent advection of the
vortex vorticity ⌶, and ␾ is the turbulent stream function, related to the turbulent vorticity in the usual way
with ⵜ 2 ␾ →r ⫺1 ⳵ / ⳵ r(r ⳵ ␾ n, ␥ / ⳵ r)⫺n 2 r ⫺2 ␾ n, ␥ ⫽ ␰ n, ␥ . The
vortex flow has been Taylor expanded about a point r 0
in the region near its edge where shear is strongest. Only
the linear variation is retained. The turbulent velocity is
u⫽⫺ⵜ ␾ ⫻z, where z is the unit vector perpendicular to
the plane of variation. The slow-scale vortex evolution
equation is
冋冕
⳵ ⌶̄ 1 ⳵
⫹
r
⳵t
r ⳵r
i⬁⫹ ␥ 0
⫺i⬁⫹ ␥ 0
d␥⬘
兺n
冉
⫺in
␾ ⫺n,⫺ ␥ ⬘ ␰ n, ␥ ⬘
r
冊册
⫽0,
(8.3)
where the integral is the inverse Laplace transform of
the turbulent vorticity flux, and the overbar on ⌶ indicates an average over the rapid time scale.
Equation (8.2) is solved by inverting the operator of
the left-hand side using a Green’s function,
␰ n, ␥ 共 r 兲 ⫽
冕
dr ⬘ G n 共 r 兩 r ⬘ 兲
⫺in
⳵⌶
␾ n, ␥
⫽0,
r⬘
⳵r⬘
(8.4)
where G n (r 兩 r ⬘ ) is the solution of
Dn
冉 冊
1 ⳵
⳵Gn
r
⫺ ␥ n G n ⫹i⍀ n⬘ 共 r⫺r ⬘ 兲 G n
r ⳵r
⳵r
⫺
n2
D G ⫽ ␦ 共 r⫺r ⬘ 兲 ,
r2 n n
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
(8.5)
and ⍀ n⬘ ⫽n(ū/r) r ⫽n ⳵ (ū/r)/ ⳵ r 兩 r 0 is the shear strain rate.
When Eq. (8.4) is substituted into Eq. (8.3), an equation
is obtained that describes the mixing of the vortex vorticity by the turbulence through an effective eddy viscosity D v :
冉
冊
⳵ ⌶̄ 1 ⳵
⳵ ⌶̄
rD v
⫽0,
⫺
⳵t
r ⳵r
⳵r
D v ⫽⫺
冕
i⬁⫹ ␥ 0
⫺⬁⫹ ␥ 0
d␥⬘
冕
dr ⬘
(8.6)
兺n
冉 冊
n2
G n共 r 兩 r ⬘ 兲
rr ⬘
⫻具 ␾ ⫺n,⫺ ␥ ⬘ 共 r 兲 ␾ n, ␥ ⬘ 共 r ⬘ 兲 典 .
(8.7)
Note that D v is an integral operator and quantities to its
right are understood to have r→r ⬘ . From Eq. (8.7) the
rate at which the vortex vorticity is mixed depends on
the turbulence level 兩 ␾ n, ␥ ⬘ 兩 2 . The Green’s function also
depends on the turbulent amplitude through the turbulent diffusivity D n , and on the shear straining of turbulence by the shear of the vortex flow through ⍀ n⬘ .
When the vortex flow has weak shear, the Green’s
function is dominated by the turbulent diffusivity and
G n ⬃D n⫺1 . This combines with the 兩 ␾ 兩 2 factor to make
the vortex mixing rate go as the eddy turnover time
( ⳵ 2 D ␯ / ⳵ r 2 ⬃nu/r⬃n 2 r ⫺2 ␾ ). In this case the vortex is
indistinguishable from a turbulent eddy and it decays,
like all eddies, on the eddy turnover time scale. When
the vortex flow has strong shear, the Green’s function is
determined by the balance of the large shear strain rate
⍀ n⬘ (r⫺r ⬘ ) and the turbulent diffusivity. As described in
Sec. II.B, this balance is achieved by a reduction of the
radial scale, yielding ␶ s ⫽ ␶ N [Eq. (2.16)] in the region of
flow shear. The Green’s function is thus proportional to
␶ s , i.e., G n ⬀⍀ n⬘ ⫺1 , giving a reduction in the magnitude
of D v . Consequently the vortex survives for many eddy
turnover times. Moreover, the radial scale of ambient
turbulence is reduced [Eq. (2.15)]. This means that diffusion of turbulence at the vortex periphery into the vortex is confined to a narrow layer whose width is given by
Eq. (2.32). These features are reflected in the solution of
D v , obtained from asymptotic boundary layer analysis
in the limit of strong shear, or ␧ s⫺1 ⫽a 3 ⍀ n⬘ /D n Ⰷ1, where
a is the radius of the vortex. The effective viscosity D v is
given by
D v⬃
冕
d␥
冉
冊
冋冉 冊
册
兺n
⫻exp
共 ␧ s⫺1 Ⰷ1 兲 ,
S⍀ n⬘ ⫺1
⫺in 2 兩 ␾ n, ␥ 兩 2
r2
共 r⫺r 0 兲 1/4共 a⫺r 0 兲 3/4
2 ⫺i⍀ n⬘
3 Dn
1/2
共 r⫺r 0 兲 3/2 ,
(8.8)
where S is a weakly varying structure function of order
unity. Due to phase mixing in summing the exponential
of a complex argument, the effective viscosity is dominated by n⫽1. Moreover, the real part of the argument
of the exponential makes D v different from zero only
within a narrow exponential layer of thickness
(D n /⍀ n⬘ ) 1/3 at the vortex edge. Stronger vortices (rela-
P. W. Terry: Suppression of turbulence by sheared flow
tive to ambient fluctuations) have a larger value of ␧ s⫺1
and therefore a smaller effective viscosity. For turbulence to mix the vortex, it must diffuse into the vortex,
extending the edge layer inward. This process is greatly
slowed by the weakness of the viscosity and its localization within a narrow layer.
The condition ␧ s ⫽D n /a 3 ⍀ n⬘ ⬍1 is dimensionally
equivalent to the condition that the vortex vorticity exceed the rms turbulent vorticity,
⌶0
⬎1,
具 ␰ 2 典 1/2
(8.9)
where the rms average is computed for an ensemble of
fluctuations in the vicinity of the vortex and ⌶ 0 is the
vortex vorticity at r⫽0. Equation (8.9) indicates that a
vorticity fluctuation becomes coherent if it lies in the tail
of the vorticity probability distribution function. For decaying turbulence, fluctuations initially in the core of the
probability distribution function, distinguished by
⌶ 0 / 具 ␰ 2 典 1/2⬍1, decay as part of the Kolmogorov cascade.
Fluctuations in the tail (⌶ 0 / 具 ␰ 2 典 1/2⬎1) decay at a far
slower rate, causing the tail probability to become enhanced with time. A Kolmogorov cascade with no coherent vortices implies a Gaussian probability distribution function. Thus an initial Gaussian distribution
function will evolve so that its core remains Gaussian
while its tail develops an enhanced non-Gaussian feature. This type of evolution is observed in intermittent
turbulence. (Interactions between two coherent vortices,
which also affect the probability distribution function,
have not been treated.)
The observed Gaussian curvature profile in the region
of a coherent vortex directly indicates that the shear
suppression criterion ␧ s ⬍1 is satisfied. The Gaussian
curvature of the vortex flow is C v ⫽r 2 n ⫺2 ⍀ n2 ⫺⌶ 2 , while
the total Gaussian curvature is
C T⫽
r 2 ⍀ n⬘ 2
n2
⫺ 共 ⌶ 2⫹ 具 ␰ 2典 兲.
(8.10)
The turbulent vorticity is included in Eq. (8.10) to account for the total squared vorticity. It is of importance
near r⫽a, where the vortex vorticity is zero. (The vanishing of vorticity at the vortex radius is implicit in the
stipulation that the vortex be localized.) The turbulent
shear stress is not included in Eq. (8.10) because it is
dominated by the vortex shear stress in the edge, and
fluctuations are small near the center. The observed
negative Gaussian curvature near the centers of the coherent vortices (McWilliams, 1984) reflects the fact that
r 2 ⍀ n⬘ 2 n ⫺2 vanishes there. At the edge, ⌶ 2 vanishes and
positive Gaussian curvature implies that a 2 ⍀ n⬘ 2 n 2 ⬎ 具 ␨ 2 典
⬇D n2 a ⫺4 , reproducing ␧ s ⬍1.
The emergence of coherent vortices in 2D decaying
Navier-Stokes turbulence thus provides an example of
shear suppression in a neutral fluid and at the same time
the elucidation of one mechanism responsible for intermittency in fluid turbulence.
B. Transport of stratospheric constituents
The atmosphere contains many trace gases and particulates, which while advected by flows as passive scaRev. Mod. Phys., Vol. 72, No. 1, January 2000
155
lars, play a crucial role in the other processes. Ozone
and chlorofluorocarbons (CFC’s) are examples. The
transport of constituents is an important factor in their
local concentrations. Significant transport occurs from
convection in large-scale circulations and flow patterns.
These flows are more coherent than turbulent, and the
associated transport and mixing can be described by a
process referred to as chaotic mixing (Pierrehumbert,
1991). Turbulence, though often not the dominant
agent, also plays an important role in constituent transport. Turbulence is responsible for flattening constituent
concentration gradients after wave-breaking instabilities
in regions descriptively called surf zones (McIntyre and
Palmer, 1984). Many observations do not resolve the
turbulent fluctuations but their presence is inferred by
other means. For example, global modeling of constituent concentrations using transport codes requires turbulent diffusivities to reproduce observed concentrations.
1. Meridional transport across zonal flows
Localized barriers to meridional (north-south) transport of constituent concentrations are commonly ascribed to regions in the stratosphere with steep concentration gradients. The gradients can arise from a variety
of processes, e.g., when rapid advection brings regions of
lowered constituent concentration into immediate proximity with regions of elevated concentration. There is a
transport barrier if the gradients persist where they
would normally be mixed by dynamical processes. Such
barriers are widely attributed to the stabilization of
wave breaking, which otherwise leads to robust transport across the gradients. Wave breaking is stabilized by
the gradient of potential vorticity, a quantity derived
from the total vorticity and the vertical stratification of
the atmosphere (Andrews, Holton, and Leovy, 1987).
Some persistent steep gradients are observed where the
flow shear is maximum and not the potential-vorticity
gradient. These maxima do not coincide because they
are proportional to first and second derivatives of the
mean flow, respectively. If there is ambient turbulence
in regions of strong shear, the shear will create a transport barrier if it is strong enough to impede turbulent
mixing across the gradient.
There is evidence for a barrier coincident with maximum flow shear that impedes the turbulent transport of
stratospheric constituents from the tropics to higher latitudes. Pronounced constituent gradients near ⫾20° latitude are evident in all seasons, not just the winter when
the potential-vorticity mechanism is in force. The location of the gradients corresponds to the edges of a zonal
flow known as the equatorial jet. The equatorial jet is a
stratospheric flow confined to tropical latitudes, generally between ⫾20°. The flow is fairly uniform between
those latitudes, and changes abruptly beyond them over
⬃5°–10°. Flow shear du/dy is therefore weak except at
⫾20°, where it is strong. The equatorial jet does not
extend into the troposphere. Contours of volcanic aerosol show poleward spreading beyond ⫾20° in the troposphere, but distinct gradients form at ⫾20° at higher el-
156
P. W. Terry: Suppression of turbulence by sheared flow
barrier to mixing and transport toward the equator.
Hartmann et al. (1989) and Proffitt et al. (1989) note that
the steep gradients occur in the region of maximum flow
shear, which is distinctly poleward of the maximum
potential-vorticity gradient (see Fig. 20 of Hartmann
et al., 1989). Similar features are evident in the data of
Tuck (1989), collected in instrumented aircraft flights
over the Arctic and Antarctic regions. In some cases
fluctuations also appear to be smaller in the region of
large flow shear (see Plate 5b of Tuck, 1989). These observations are anecdotal but suggestive, indicating that
further analysis is warranted.
2. ␤-plane model of meridional transport
FIG. 16. Contours of Gaussian curvature for coherent vortices
in simulations of decaying two-dimensional Navier-Stokes turbulence. The vortices consist of a central region of negative
Gaussian curvature surrounded by a region of positive Gaussian curvature. Away from the vortices, the value of the Gaussian curvature fluctuates near zero. From McWilliams, 1984.
evations in the stratosphere (Trepte and Hitchman,
1992; Hitchman, McKay, and Trepte, 1994). The effect is
particularly striking when large quantities of aerosols
are injected into the atmosphere after a major volcanic
eruption in the tropics. Aerosols injected into the stratosphere by Mt. Pinatubo in June of 1991 spread rapidly
poleward, within a band between ⫾20° and along the
equator. Figure 17 shows the global distribution of
stratospheric aerosols a little more than a month after
eruption. A near homogenization of aerosol concentrations in the equatorial direction is already evident. Concentrations are nearly uniform within the tropical band
between ⫾20° and have steep gradients northward and
southward. The persistence of these gradients is illustrated in the temporal record of aerosol concentrations
following the eruption of Nevado del Ruiz in November
of 1985. Figure 18 shows that these gradients persisted
for four years after the eruption. (Eruptions of Mt. Kelut in 1990 and Mt. Pinatubo in 1991, which saturated
the instruments, appear in the latter part of the record.)
The steep concentration gradients closely coincide with
the zone of steepest flow shear, as indicated in Fig. 6 of
Trepte, Veiga, and McCormick (1993).
Zonal flows known as the Arctic and Antarctic winter
polar vortices occur over the wintertime poles. These
vortices extend from the poles to approximately 60°,
with a zone of strong shear at the edge. The catalytic
ozone depletion cycle of late winter leads to reduced
and elevated concentrations of O3 and ClO, respectively,
in the polar regions. These concentrations are quite uniform over the poles, implying mixing, but have sharp
gradients at the edge of the polar vortex, suggesting a
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
Transport in situations like those of the prior subsection has been studied numerically with the quasigeostrophic ␤-plane model [Eq. (3.20)] (Shepherd, 1987;
Ware et al., 1995, 1999). In these studies a mean zonal
flow with meridional shear was specified as a model for
flows such as the equatorial jet and polar winter vortices.
For computational simplicity, the mean flow profile was
a one-period sinusoid, allowing periodic boundary conditions in both zonal and meridional directions. The turbulence was initialized and allowed to decay, or driven
by an external force. In the simulations, turbulence is
not driven by the mean flow because ␤ is chosen sufficiently large relative to the second derivative of the
mean flow to make the flow barotropically stable, as represented in the condition of Eq. (3.4). The largest scales
of the turbulence were smaller than the meridional scale
of the mean flow by a factor of ⬃5. This turbulence is
representative of fluctuations created by wave breaking
or driven by other processes, such as waves propagating
upward from the troposphere.
Shepherd (1987) observed a reduction in turbulent
vorticity where the zonal mean flow had maximum
shear, but provided no explanation. The shear suppression criterion ␧ s ⬍1 was not a part of the analysis or
interpretation, and it is difficult to determine how
strongly it was satisfied. In Ware et al. (1999), parameters were chosen to make ␧ s as small as 0.1 in the region of maximum shear. Figure 19 shows the mean flow
profile and values of ␧ s across the jet. (Large fluctuations of ␧ s arise from scatter in the turbulent correlation
time.) The vorticity contours are shown in Fig. 20(a).
There is a near absence of turbulence where ␧ s ⬍1 and
strong turbulence elsewhere. For comparison, Fig. 20(b)
shows vorticity contours for a case with no jet (␧ s
→⬁). There the turbulence is isotropic as expected. Figures 19 and 20 represent steady turbulence driven by an
external force. Similar results were reported for damped
turbulence. Ware et al. (1995, 1999) also examined passive scalar transport using tracer particles and calculating an effective spatial diffusivity from the separation of
pairs of tracer particles as a function of time. Consistent
with Fig. 20(a), the diffusivity was minimum where ␧ s
was minimum. The diffusivity decreased with jet amplitude for fixed mean flow scale length (⬃␧ s⫺1 ). Above a
critical amplitude, however, the jet became unstable
P. W. Terry: Suppression of turbulence by sheared flow
157
FIG. 17. Distribution of stratospheric aerosols approximately
one month after the eruption of
Mt. Pinatubo. Aerosol concentration is nearly homogeneous
along the equator with sharp
gradients around ⫾20° latitude.
From Trepte, 1993 [Color].
FIG. 18. Zonal averages of aerosol concentration at two altitudes in the stratosphere during the years 1985–1991. Large increases associated with the eruption of Nevado del Ruiz
in November of 1985 persisted for years
within the latitudes bounded by ⫾20° but did
not spread poleward beyond these latitudes in
any significant way. From Trepte, 1993
[Color].
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
158
P. W. Terry: Suppression of turbulence by sheared flow
( ⳵ 2 ū/ ⳵ y 2 ⬎ ␤ ), and the diffusivity rose abruptly. Above
the instability threshold diffusivities in the zonal and
meridional directions were comparable, while below the
threshold the meridional diffusivity was much smaller
than the zonal diffusivity.
C. Shear flow in self-organized criticality
The suppression of transport by flow shear has been
observed in systems with self-organized criticality (SOC;
Newman et al., 1996). In such systems, a local flux induces transport at adjacent locations, leading in some
cases to extended avalanche-like transport events whose
correlation length and time exceed those of the fluctuation spectrum. Systems with self-organized criticality are
not limited to flows, but have been used to model transport in granular media such as sandpiles (Bak, Tang, and
Weisenfield, 1987, Nagel, 1992), networks, and even
seismic faults. By adopting appropriate generalizations
for flow shear, systems with self-organized criticality
have provided the greatest potential to date for application of the notion of shear suppression to dynamical systems.
The paradigm of self-organized criticality was introduced to plasma physics to model the global transport
caused by small-scale fluctuations in terms of a universal
process transcending particular collective instabilities
(Diamond and Hahm, 1995). In these models, transport
organizes itself on a hierarchy of spatial scales, ranging
from the fluctuation scale, which may be limited by the
magnetic field to a gyroradius, to the system size. The
organizing principle is a ‘‘bucket-brigade’’ effect linking
discrete cells of spatially localized fluctuation activity.
The arrival of transported material at a cell location induces local instability by pushing the gradient above its
local instability threshold. The resulting instability transports the incident material to the next cell. If the local
instability in each cell transports more than the incident
material, the chain of transport events can grow in space
and time and can become global in nature. Such largescale avalanche-like events lead to a characteristic frequency spectrum subrange with a power-law decay of
f ⫺1 . The fact that individual cells transport more than
the incident material leads to a profile that is subcritical,
i.e., the gradient is below the instability threshold, except during sporadic transport events.
A model realization of self-organized criticality is the
sandpile automaton (Bak, Tang, and Weisenfield, 1987;
Nagel, 1992). Transport of sand down the pile is forced
by grains of sand randomly falling from above. When
the sandpile gradient becomes too steep locally, sand
slides down the pile. The process is modeled using a grid
to track the amount of sand in each cell. At each time
step, if the difference in the amount of sand between
each cell and its neighbor exceeds a threshold, a fixed
amount of it is removed from the cell and transported to
the neighboring site. If the difference does not exceed
the critical threshold, no sand is transported. The process is governed by the simple prescription:
If Z n 共 t 兲 ⭓Z crit
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
h n 共 t⫹1 兲 ⫽h n 共 t 兲 ⫺N f ,
h n⫹1 共 t⫹1 兲 ⫽h n⫹1 共 t 兲 ⫹N f ,
(8.11)
where h n (t) is the amount of sand in the nth cell at time
t, Z n (t)⫽h n (t)⫺h n⫹1 (t) is the local gradient, Z crit is the
critical gradient, and N f is the amount of sand transported between adjacent cells in one time unit. A
sample parameter set is Z crit⫽8, N f ⫽3, and Z n (t 0 )⫽7.
If a grain of sand (one unit) is added to site n, the
threshold gradient is exceeded and three grains of sand
are passed to the next site. If adjacent sites are also near
threshold, it is apparent that sites n⫺1 and n⫹1 could
exceed the threshold at the next time step. In this way a
single grain of externally incident sand can initiate a
transport event that unfolds over many time steps and
affects many sites both uphill and downhill from the initial site. On the other hand, if the same sample parameter set applies to a site n whose neighboring sites n
⫺1 and n⫹1 are well below the threshold, the removal
of three grains from site n to n⫹1 may be the extent of
the event. There is therefore a hierarchy of events, from
those that affect only adjacent sites to those that affect
the entire sandpile.
Shear flow can be introduced into the sandpile automaton as a flow transverse to the sandpile gradient
with shear in the gradient direction (Newman et al.,
1996). The spatial grid of cell sites must now be two
dimensional, describing, for example, a conical sandpile
on a planar surface. At the top and bottom of the sand
pile the sand flows at constant speed transverse to the
gradient in opposite directions. Connecting these flows
is an intermediate region where the flow is sheared,
matching the oppositely directed flows at top and bottom, and passing through zero in the middle. The flow is
added to the dynamics in the time advance step after
moving any falling grains to their new positions. The
sheared flow is observed to dramatically decrease the
number of large-scale transport events, i.e., events involving sites across the entire shear zone. If the incident
sand flux is the same for sandpiles with and without flow,
the net transport of sand over time will be the same in
both cases. Consequently in the sheared case there is an
increase in small-scale events to compensate for the decrease in large-scale events. The diffusivity changes
functional form as a result of the flow shear, although
there is not a marked decrease in its magnitude. The
mechanism for the decrease of large-scale events is the
simple decorrelation of these events by the flow shear.
IX. CONCLUSIONS
A. Summary
The suppression of turbulence and turbulent transport
by stable flow shear is a robust process in plasmas and
neutral fluids. The physical mechanism is the acceleration of turbulent decorrelation by mean flow shear, and
it is in force whenever ␶ s ⬍ ␶ N ⬍ ␶ D , where ␶ s , ␶ N , and
␶ D are the shear straining time, turbulent correlation
time, and domain time, respectively. This hierarchy of
P. W. Terry: Suppression of turbulence by sheared flow
FIG. 19. Meridional profiles of a zonally averaged zonal flow
and the shear suppression parameter ␧ S in a simulation of geostrophic turbulence in a beta plane. According to the predictions of Sec. II, when the shear suppression parameter is less
than unity, as it is in the regions of maximum flow shear, turbulence is suppressed. From Ware et al., 1999.
time scales distinguishes the physics of shear suppression
from rapid-distortion theory, which holds when ␶ s ⭐ ␶ D
⬍ ␶ N . With the domain time shorter than the turbulent
correlation time in rapid-distortion theory the dynamical
equations can be linearized. The distinction between the
theory of shear suppression and rapid-distortion theory
is apparent in the analysis of the Fourier-Laplace expansion of the inverted advective operator for turbulence in
a simple background flow with unidirectional plane
shearing (Sec. II.B.3). These two theories emerge as the
long-time and short-time asymptotic limits. As such the
theory of shear suppression is a nonlinear extension of
rapid-distortion theory. The dynamical importance of
the turbulent correlation time in the theory of shear suppression causes the turbulence to adjust itself (as in
slowly changing turbulence) to bring the shear straining
time and correlation time into balance, ␶ s ⫽ ␶ N . This is
accomplished by a reduction in the cross-shear scale
length to make ␶ N as small as ␶ s . If the driving source
remains invariant across the shear flow, the decrease in
␶ N directly leads to a decrease in the turbulent energy,
hence to a suppression of turbulence.
The suppression of turbulence in shear flows has been
observed in plasma experiments and in simulations of
magnetically confined plasmas. It is a robust property of
analytic theory ranging from simple dimensional scaling
analysis to more rigorous asymptotic theory. As a dynamic phenomenon in fusion plasmas, this process is unusually universal. It occurs in regions of the plasma
ranging from the edge to the core. These are regions in
which the turbulence dramatically changes character because of different driving mechanisms and parameters,
such as temperature (which can range over three orders
of magnitude). This process also occurs in numerous
plasma confinement devices, with widely varying geometries, magnetic topologies, and plasma conditions. The
flow shear can arise from a variety of mechanisms, exRev. Mod. Phys., Vol. 72, No. 1, January 2000
159
FIG. 20. Contours of constant vorticity in simulations of externally driven beta-plane turbulence, (a) with no mean zonal
flow, and (b) with the mean zonal flow of Fig. 19. Suppression
of the turbulence in the regions where ␧ s ⬍1 is clearly evident.
From Ware et al., 1999.
ternal and internal. Moreover, shear suppression occurs
in hydrodynamics and in more general transporting systems with self-organized criticality.
The use of flow shear to create transport barriers in
fusion plasmas represents a major breakthrough in the
ability to achieve fusion in magnetically confined plasmas. The seemingly irreducible transport associated with
small-scale turbulence, driven by the large gradients of
temperature and density in fusion plasmas, has long
been an impediment to magnetic fusion. With flow
shear, transport in some cases has been reduced to the
absolute minimum associated with Coulomb collisions
that occur because of thermal motion. A variety of
present-day records have been set in magnetic fusion
devices—for fusion power produced, fusion power efficiency, and confinement time—utilizing flow-shearinduced transport barriers.
The effect of flow shear in plasmas is not limited to
the way it accelerates the nonlinear decorrelation process. Though the mechanisms are less general, flow
shear can stabilize a variety of collective instabilities in
fusion plasmas. Flow shear affects the complex phase
angle between an advected fluctuation and the advecting
flow. This produces a reduction factor in the transport
flux that is independent of amplitude reduction. Flow
shear also disrupts large-scale transport events in systems with self-organized criticality.
Shear flow is generated in plasmas by a variety of
mechanisms in what is often a very complicated fashion.
The radial electric field required for the E⫻B flow can
be driven externally by probes or internally by the pressure gradient, by ion flows in the toroidal and poloidal
directions, or by differential charge loss. The ion flows,
in turn, are driven by the Reynolds stress, by injected rf
waves, or by asymmetries in the transport flux. The creation of shear flows in plasmas by internal mechanisms
that are themselves modified by the flow shear and its
associated reduction in transport has led to a new picture of the magnetically confined plasma as a selfregulated state. Shear suppression gives this state the
curious feature that increases in free energy, in the form
of greater external heat input, can lead to lower turbu-
160
P. W. Terry: Suppression of turbulence by sheared flow
lence and steeper gradients, rather than the opposite.
The complicated nonlinear connections between internally generated flow shear, turbulence, and profiles of
the mean quantities that drive turbulence and transport
lead to bifurcations of the plasma state, the most universal being the transition from the L to the H mode.
At present, studies of shear suppression outside
plasma physics have been limited but promising. These
include a demonstration that flow shear in the edge of
intense vortices in Navier-Stokes turbulence allows
these vortices to greatly reduce or effectively eliminate
the mixing of vorticity by ambient turbulence. This process provides an explanation for intermittency in decaying 2D Navier-Stokes turbulence and explains in a natural way the observed evolution of probability
distribution functions and the Gaussian curvature. Shear
suppression also affects geostrophic turbulence and its
transport of constituents in the stratosphere. The observed presence of transport barriers in regions of strong
flow shear at the edges of the equatorial jet and polar
winter vortices may be associated with this mechanism.
Systems with self-organized criticality are more general
than the hydrodynamic flows of the Navier-Stokes equations and the related atmospheric turbulence models.
The existence of a form of shear suppression in these
systems, affecting the large-scale correlated transport
events, represents an application to the broadest class of
dynamical systems considered to date.
B. Open questions
This review has examined the suppression of turbulence by flow shear, assuming flows that are stable, 2D,
and steady on time scales that are long compared to the
turbulent correlation time. What occurs under other
conditions is, in many cases, an open question. For example, unstable shear flows drive turbulence rather than
suppress it. However, if the instability is intermittent,
driving and suppression may occur intermittently, leading to some reduction of turbulent energy in a timeaverage sense relative to a steady instability process.
This situation ultimately applies to the stratosphere
when the mean flow is periodically unstable to wavebreaking events, although if the time between wavebreaking events is longer than the turbulent correlation
time, the flow can be treated as stable. Intermittent stability may also account for the reduction of turbulence
observed in certain 3D atmospheric boundary-layer
flows (Smedman, Bergström, and Högström, 1995). In
general, 3D dynamics complicate the picture presented
in this review. While flow shear may accelerate the decorrelation of eddies whose vorticity is not aligned with the
flow, leading to a suppression of vorticity, aligned eddies
are stretched and their vorticity is amplified. Alignment
(or lack thereof) is itself a product of the turbulence,
making it difficult to anticipate what role shear suppression ultimately plays in 3D shear flows.
Of recent interest, unsteady zonal flows (i.e., toroidally and poloidally symmetric flows) arise in simulations of ion-temperature-gradient turbulence in tokaRev. Mod. Phys., Vol. 72, No. 1, January 2000
maks (Sec. V.A.2) and are observed to lead to lower
turbulent energy and transport (Lin et al., 1998). Although this effect has been attributed to a suppression of
turbulence by the shear in the zonal flows (Diamond
et al., 1998), the zonal flows are incoherent and appear
in simulations to decorrelate on a time scale that is comparable to that of the turbulence. Other effects may also
play a role, such as diminished efficiency of spectral
transfer with zonal flows.
The suppression of turbulence by flow shear in plasmas generally involves turbulence excited by a collective
instability. It is relatively easy to determine the effect of
flow shear on the linear stability. How it affects saturated turbulence in an inhomogeneous magnetic field is
more difficult to assess because the saturated state may
involve nonlinear changes in the eigenmode structure.
The nature of shear suppression in externally driven turbulence has not been explored beyond simple dimensional analysis. It was assumed in Sec. II.B that flowwise scales are not affected by the flow shear. The
validity of this assumption depends on how incompressibility is maintained under the forcing, an issue that has
not been examined.
There is a wide consensus within the magnetic fusion
research community that flow shear is responsible for
the transport barriers observed in experiment. At
present there is no consensus on the mechanism responsible for generation of the shear flow. Despite the large
number of mechanisms described in Sec. IV, none has
been shown to be invalid or inoperative. This statement
even applies to externally driven flow shear, where although there is clearly an external current as described
in Secs. IV.B and VII.B.1, there are discrepancies between the observational details and model predictions,
and the measurements do not account for all the forces,
e.g., the Reynolds stress. Considerable work remains to
be done, especially in measuring quantities of relevance
to the proposed mechanisms and in producing predictive
(as opposed to heuristic) models of flow generation. The
same statements also apply to transition modeling.
Again there are a large number of models (Sec. VII.B),
none of which has been shown to be invalid.
Open questions remain concerning the details of
transport suppression in fusion experiments. For example, in the internal transport barrier, the suppression
of the ion heat and particle fluxes appears to be stronger
in many cases than the suppression of the electron heat
fluxes. This difference is not understood. Moreover, certain experiments, particularly those with probe-induced
H modes, have temperature fluctuation and heat flux
reductions with flow shear, while similar experiments in
other tokamaks do not. This difference may be accounted for by the variation of the parallel thermal conduction from device to device, but has yet to be demonstrated. The existence of transport barriers in plasmas
necessarily involves a reduction in plasma transport. Yet
most theoretical work on shear suppression treats only
the amplitude reduction, and not the effect of flow shear
on the cross phase (Sec. V.B). Theories that describe the
latter have been restricted to a weak-flow-shear regime,
P. W. Terry: Suppression of turbulence by sheared flow
which is not likely the regime of experiments. While
flow shear has been shown to reduce the cross phase, a
number of experimental observations relating to the
cross phase have yet to be explained. These include observations of a reduction of the cross phase with little or
no accompanying reduction of amplitudes and observations that fluxes can change sign in transport barriers.
C. Future directions
Despite the widely held view that flow shear is responsible for the transport barriers observed in fusion experiments, the evidence is inferred in experiments with
highly complex dynamics, complicated geometries, and
many variables, few of which can be externally controlled. A simple, direct demonstration of shear suppression, ideally in a controlled neutral-fluid experiment, is a
desirable direction for future work. In a neutral fluid,
the number of collective modes is limited, there is a
single unequivocal flow velocity (as opposed to a plasma
in which each charge species has its own flow and flow
dynamics), and the geometry can be simple because it
does not have to confine an ionized gas. The constraints
of a stable shear flow and 2D turbulent dynamics could
be realized in a rotating soap film with grid-generated
turbulence or in a differentially rotating tank with a turbulent Rayleigh Bénard flow. Such an experiment could
not only verify the general mechanism of shear suppression, but measure scalings and spectra, and examine the
effect of different forcing mechanisms.
Much work remains to be done in measuring quantities of importance to shear suppression in fusion experiments. This is particularly true of the Reynolds stress.
This paper has presented numerous parallels between
plasmas and neutral fluids, which in many cases lead to
strikingly similar expressions and effects. In neutral fluids the Reynolds stress not only leads to turbulent viscosities, but it generates a number of geophysical flows
such as the quasibiennial oscillation (Holton and
Lindzen, 1972; Plumb, 1977). The importance of the
Reynolds stress as a flow generation mechanism in neutral fluids suggests that it should not be ignored in
plasma turbulence. Indeed, all of the measurements of
the Reynolds stress in plasmas have indicated that its
magnitude is sufficient for it to contribute to observed
flows. Measurement of the Reynolds stress in fusion
plasmas is very difficult but is crucial to a detailed understanding of transport barrier physics.
Shear suppression has the potential for leading to future fusion reactors whose size is manageable and whose
costs are competitive with other nonfossil forms of future energy generation. Although record confinement
parameters have been achieved with flow-shear-induced
transport barriers, these barriers are transient. The internal transport barrier requires a hollow current profile,
which is not maintained in a steady state because of inward current diffusion. Moreover, the confinement is so
good in these discharges that pressure gradients steepen
to the point where they can become unstable to global,
MHD instabilities that can terminate the discharge. The
Rev. Mod. Phys., Vol. 72, No. 1, January 2000
161
development of techniques to produce steady and controllable shear flow is crucial if flow shear suppression is
to provide the confinement needed in future steady-state
fusion reactors. A better understanding is also needed of
the requirements of flow generation in reactor regimes,
the availability of enhanced-confinement modes in reactors, and the scaling of transition thresholds to reactor
parameters.
Finally, some speculations are offered for non plasma
situations, other than those described in Sec. VIII, in
which shear suppression might apply. The Earth’s turbulent boundary layer has already been mentioned in connection with observations of suppression when a jet
forms near a land/sea interface (Smedman, Bergström,
and Högström, 1995). The process of shear suppression
may apply to wall flows during the transient period of
laminarity after bursts of turbulent eddy formation at
the wall. A form of shear suppression has already been
invoked in the turbulent boundary layer above an undulating surface (Hunt, Leibovich, and Richards, 1988).
The ocean has phenomena that possibly relate to shear
suppression. These include gulf stream rings and Mediterranean salt lenses (meddies). Both are long-lived coherent vortical flows that sequester elevated concentrations of heat and, in the case of meddies, salinity. These
concentrations are subject to outward diffusion by ambient turbulence and mixing and their longevity may
suggest a transport barrier. Astrophysics may also
present situations in which shear suppression applies,
particularly in relationship to the differential rotation of
Keplerian motion.
ACKNOWLEDGMENTS
The author acknowledges helpful conversations with
J. A. Boedo, K. Burrell, J. D. Callen, B. A. Carreras, B.
E. Chapman, P. H. Diamond, P. Durbin, E. Fernandez,
G. Fiksel, C. Forest, R. Gatto, T. S. Hahm, A. B. Hassam, C. C. Hegna, M. H. Hitchman, R. A. Moyer, P.
Moin, D. E. Newman, S. C. Prager, W. C. Reynolds, J.
Sarff, L. Smith, A. S. Ware, and S. Zweben. This work
was supported by the U.S. Department of Energy under
Grant No. DE-FG02-89ER-53291 and by the Vilas
Trustees.
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